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Classical (non-racial) systems of slavery in Western civilization, such as Greco-Roman, Celtic, etc. typically included provisions for selling oneself into slavery. This might be as a way to escape crushing debt, or done as a matter of honor to serve a benefactor. These methods existed in addition to other ways that one might become a slave, such as being taken as a Prisoner Of War or being convicted of certain criminal offenses.

Did this principle apply, at least in theory, to slavery as practiced in the USA before the Thirteenth Amendment? I recognize that there were debt-based servitude arrangements not amounting to total, lifetime enslavement (e.g. Indentured Servitude) that were used primarily by white immigrants, but was legally enslaving a free, willing US resident or immigrant legally impossible by any means or was this simply something that was not done for social or practical reasons?

If "The United States" is too broad for such an analysis, we can limit ourselves to the state of Virginia, which had some of the earliest legislation surrounding the "peculiar institution".

For the purposes of this question, colonial-era law (pre-independence) may count.

This question has nothing to do with unlawful human trafficking, nor does it have anything to do with BDSM-culture "slave contracts" and similar phenomena.

In British colonies and their successors

It is hard to prove a negative, but in the British colonies and their successors, I believe that the answer was probably no.

Indentured servitude came close to the concept of selling oneself into slavery, although for a fixed term of years. And, the 13th Amendment recognizes the possibility that slavery could be a punishment for a criminal offense, although I've never heard of any instances in which this actually was a punishment for a criminal offense (although the equivalent of indentured servitude was a punishment in some cases) outside on case from 1659 in what is now New Mexico but then was part of Old Mexico, mentioned below.

I've also never heard of even a single instance of someone selling themselves into slavery in the U.S., despite being a fairly avid history buff who minored in history in college.

Another piece of circumstantial evidence that argues against the existence of a mechanism for selling yourself into slavery is the fact that the U.S. Constitution, all the way back in 1789, already provided for, as a basic function of government, for bankruptcy, and the institution of bankruptcy was one with long British law antecedents that existed in the American colonies as well. Likewise, the U.S. Constitution also prohibited punishments involving a "corruption of blood" which held descendants responsible for the acts of their ancestors even if those acts amounted to treason.

Self-enslavement for "honor" is very akin to a feudal oath of fealty, which was strongly rejected as part of the Independence movement of the United States including language in the 1789 U.S. Constitution, and it was also an institution that never had much currency in the Americas because many of the early colonies were of a corporate or religious nature, rather than an aristocratic one, unlike many early Spanish colonies.

These historical circumstances eliminated the main circumstances motivating historical examples of an institution of submitting oneself to slavery.

Another historical fact that strongly argues against there being an institution of selling oneself in slavery was the strongly racialized nature of slavery in British colonies and their successors. Indentured servitude was for white people, slavery was for black people, and there really wasn't much of a cultural or conceptual need for mix up these two models. Doing so would have undermined a cultural axiom of racial supremacy which was important to sustaining slavery as an institution in the United States.

This may also be a reason that there was not a strong institution of enslaving either European descent or Native American prisoners of war in North America, even though some Native Americans were enslaved on a more ad hoc basis by people of European descent. If I recall correctly, early attempts to enslave Native Americans were also not very successful as an economic proposition for the would be slavers, for whatever reasons, further racializing slavery as an institution in North America largely limited to people of African descent. The discussion at the link above illustrates the parameters of the situation:

When Europeans arrived as colonists in North America, Native Americans
changed their practice of slavery dramatically. Native Americans began
selling war captives to Europeans rather than integrating them into
their own societies as they had done before. As the demand for labor
in the West Indies grew with the cultivation of sugar cane, Europeans
enslaved Native Americans for the Thirteen Colonies, and some were
exported to the "sugar islands." The British settlers, especially
those in the southern colonies, purchased or captured Native Americans
to use as forced labor in cultivating tobacco, rice, and indigo.
Accurate records of the numbers enslaved do not exist. Scholars
estimate tens of thousands of Native Americans may have been enslaved
by the Europeans, being sold by Native Americans themselves or
European men.

Slaves became a caste of people who were foreign to the English
(Native Americans, Africans and their descendants) and non-Christians.
The Virginia General Assembly defined some terms of slavery in 1705:

All servants imported and brought into the Country ... who were not
Christians in their native Country ... shall be accounted and be
slaves. All Negro, mulatto and Indian slaves within this dominion ...
shall be held to be real estate. If any slave resists his master ...
correcting such slave, and shall happen to be killed in such
correction ... the master shall be free of all punishment ... as if
such accident never happened.

— Virginia General Assembly declaration, 1705.

The slave trade of Native Americans lasted only until around 1730. It
gave rise to a series of devastating wars among the tribes, including
the Yamasee War. The Indian Wars of the early 18th century, combined
with the increasing importation of African slaves, effectively ended
the Native American slave trade by 1750. Colonists found that Native
American slaves could easily escape, as they knew the country. The
wars cost the lives of numerous colonial slave traders and disrupted
their early societies. The remaining Native American groups banded
together to face the Europeans from a position of strength. Many
surviving Native American peoples of the southeast strengthened their
loose coalitions of language groups and joined confederacies such as
the Choctaw, the Creek, and the Catawba for protection.

Native American women were at risk for rape whether they were enslaved
or not; during the early colonial years, settlers were
disproportionately male. They turned to Native women for sexual
relationships. Both Native American and African enslaved women
suffered rape and sexual harassment by male slaveholders and other
white men.

The exact number of Native Americans who were enslaved is unknown
because vital statistics and census reports were at best
infrequent. Andrés Reséndez estimates that between 147,000 and
340,000 Native Americans were enslaved in North America, excluding
Mexico. Linford Fisher's estimates 2.5 million to 5.5 million Natives
enslaved in the entire Americas. Even though records became more
reliable in the later colonial period, Native American slaves received
little to no mention, or they were classed with African slaves with no
distinction. For example, in the case of "Sarah Chauqum of Rhode
Island", her master listed her as mulatto in the bill of sale to
Edward Robinson, but she won her freedom by asserting her Narragansett
identity.

Little is known about Native Americans that were forced into labor.
Two myths have complicated the history of Native American slavery:
that Native Americans were undesirable as servants, and that Native
Americans were exterminated or pushed out after King Philip's War. The
precise legal status for some Native Americans is at times difficult
to establish, as involuntary servitude and slavery were poorly defined
in 17th-century British America. Some masters asserted ownership over
the children of Native American servants, seeking to turn them into
slaves. The historical uniqueness of slavery in America is that
European settlers drew a rigid line between insiders, "people like
themselves who could never be enslaved", and nonwhite outsiders,
"mostly Africans and Native Americans who could be enslaved". A unique
feature between natives and colonists was that colonists gradually
asserted sovereignty over the native inhabitants during the
seventeenth century, ironically transforming them into subjects with
collective rights and privileges that Africans could not enjoy. The
West Indies developed as plantation societies prior to the Chesapeake
Bay region and had a demand for labor.

In the Spanish colonies, the church assigned Spanish surnames to
Native Americans and recorded them as servants rather than slaves.
Many members of Native American tribes in the Western United States
were taken for life as slaves. In some cases, courts served as
conduits for enslavement of Indians, as evidenced by the enslavement
of the Hopi man Juan Suñi in 1659 by a court in Santa Fe for theft of
food and trinkets from the governor's mansion. In the East, Native
Americans were recorded as slaves.

The colonial legislation from Virginia from 1705 quoted above, in particular, argues that Europeans could not generally be enslaved in Virginia, although the distinction was formally religious rather than racial and hence, could have authorized Jewish slavery in Virginia (there were almost no Jews in England in the colonial era because they were exiled from Britain at one point). The religious aspect also had a caveat omitted in the quote from the 1705 legislation from Virginia above:

all servants imported and brought into the Country... who were not
christians in their native country, (except... Turks and Moors in
amity with her majesty, and others that can make due proof of their
being free in England, or any other christian country, before they
were shipped...) shall be accounted and be slaves, and such be here
bought and sold notwithstanding a conversion to christianity
afterward.

Also not unrelated is the fact that early attempts to use indentured servants and hired farm hands of European descent in the American Southeast were not very successful because the mortality and disease rates of these servants, especially due to mosquito born diseases to which they had little natural immunity, was so significantly greater than that of African descent slaves who had some natural immunity to these diseases, making white slavery an economically unattractive institution to establish.

Periodic rounds of "moral panic" related to the nebulous and often not very well grounded fear of white slavery that has recurred throughout U.S. history at regular intervals also argues for the racialized nature of the institution in the U.S. and its predecessor British colonies.

In sum, there is very good reason to believe that it was never possible to sell oneself into slavery in the United States or in the British colonies that preceded it, even though it is hard to prove that point definitively.

was legally enslaving a free, willing US resident or immigrant legally
impossible by any means or was this simply something that was not done
for social or practical reasons?

I do not believe that this was expressly provided for by statute, but most law at the time was common law rooted in British case law anyway, which makes distinguishing between something that was "legally impossible" and something "that was not done for social or practical reasons" harder to distinguish in common law jurisdictions than in civil law jurisdictions based upon continental Europe.

Colonies of Civil Law Countries

I am less confident of the status of this possibility in the case of Dutch or Spanish or French colonies in North America.

I know from historical accounts of French North America that slavery, while it existed, was much less racialized with a significant community of free people of color.

And, all three of those countries had legal systems with their roots in Roman law that was "received" by these countries in the early modern era basically when political leaders as the Middle Ages started to fade away, started to use Roman legal treatises to justify their resolutions of disputes without formally adopting this as a positive source of law (the Netherlands is a bit more muddled as it was also a major participant in the home grown institution called the "law merchant" devised by merchants serving as arbitrators, especially in disputes between merchants, some of which ended up influencing British common law). Anyway, in these countries, Roman legal sources that recognized the concept of selling yourself into slavery would have been available and could have been considered good law.

Also, because these countries, especially Spain and France, had much less of a vibrant commercial tradition in the early modern period, it is quite likely that the institution of bankruptcy that developed comparatively early in Britain came later there, potentially necessitating a way to settle debts that could not be discharged and which might otherwise burden descendants (many places in continental Europe had descendants liable for the debts of their ancestors into the early modern period).

So, if it were ever possible to sell oneself into slavery at all in North American history, in what ultimately became the United States, it probably happened in French North America, former Spanish colonies, or in Dutch Manhattan, with the first two probably much more likely to have occurred than the last one.

Even there, however, I cannot think of a single historical or literary account that recognizes even one example of such a transition in personal status. A short account of French slavery law also provides no precedent for anything remotely similar to this practice. This is particularly notable since the French apparently largely borrowed from Dutch and Spanish practice.

In Pre-Colonial America

The possibility of selling oneself into slavery did exist in the legal and cultural circumstances of some Native American tribes in North America, where the institution was less like the chattel slavery of the early English slave traders. As explained at the link above:

Many Native American tribes practiced some form of slavery before the
European introduction of African slavery into North America.

Native American groups often enslaved war captives whom they primarily
used for small-scale labor. Others however would stake themselves in
gambling situations when they had nothing else which would put them
into servitude for a short time in some cases for life; captives were
also sometimes tortured as part of religious rites, and these
sometimes involved ritual cannibalism. During times of famine some
Native Americans would also temporarily sell their children to obtain
food... Several tribes held captives as hostages for payment. Various
tribes also practiced debt slavery or imposed slavery on tribal
members who had committed crimes; full tribal status would be restored
as the enslaved worked off their obligations to the tribal society.
Obtaining prisoners was also a strong interest for Native American
warriors as for the qualification of being considered brave this was
especially an interest of male warriors in various tribes. Other
slave-owning tribes of North America included Comanche of Texas, the
Creek of Georgia; the fishing societies, such as the Yurok, who lived
in Northern California; the Pawnee, and the Klamath.

Self-enslavement seems to go hand in hand with slavery not having a strong racial component.