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The First Amendment to the US Constitution reads:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the
freedom of speech, or of the press; or the right of the people
peaceably to assemble, and to petition the Government for a redress of
grievances. (Emphasis Mine)

Since the Amendment begins with the clause "Congress shall make no law...," does that mean the president is free to pass executive orders infringing on the specific rights enumerated? I'm reading it as as long as Congress doesn't pass the law, then violations are permissible, subject only to impeachment.

The reason this isn't a loophole is that The President doesn't have the power to make laws at all, only Congress can do that. Thanks to the vast expansion in the power of the Executive branch in recent decades, Executive Orders certainly seem like laws which The President can make on their own. Legally, though, they are actually just instructions to employees of the Executive Branch on how to interpret laws passed by Congress.

So, while The President could attempt to use Executive Orders to limit Constitutional rights, the authority of that order would be rooted, however tenuously, in a law passed by Congress – and it could be found to violate the Constitution by the courts.

The U.S. Supreme Court has held that all executive orders from the president of the United States must be supported by the Constitution, whether from a clause granting specific power, or by Congress delegating such to the executive branch. Specifically, such orders must be rooted in Article II of the US Constitution or enacted by the congress in statutes.

https://en.wikipedia.org/wiki/Executive_order#Basis_in_the_United_States_Constitution

The president can always try (so can Congress). But the problem is that the Supreme Court can always rule against the order if and when it becomes contested in court. And if the order blatantly violates freedom of speech, then the Supreme Court will almost certainly strike it down.

In other words, the Supreme Court is the final judge of whether a law/order is unconstitutional. Let me add that the Supreme Court also decides whether or not this applies to the president's orders.

What happens then if the Supreme Court is packed completely full of flunkies to the president? Then theoretically the Court might let the president do whatever he wants. But I think that's unlikely to happen, because any SC that did this would quickly lose any legitimacy as a court. And Justices care a lot about their legitimacy, because that's all the authority they have.

One of the less discussed exceptions to the First Amendment is that the Government may restrict speech in it's capacity as an employer as if it was a private employer. That is to say that employees of the Executive Branch (the largest segment of federal employees, given that this includes the U.S. Military) are allowed to make reasonable restrictions on workplace speech and not violate the first amendment. Typically these are only enforceable when the speech is made while on duty or acting as a representative of the government. The military is highly restricted as they are technically 24/7 duty of all individuals.

The Hatch Act is a good guide for some of the restrictions on speech that executive employees can not take part in (mostly it relates to political campaigns) and binds different departments differently. As a general rule, these bindings apply only to employees of the executive branch of government who are not elected (The President and Vice President) or directly appointed officers of the executive (i.e. any post that requires advice and consent of the United States Senate) as those roles are inherently political. Political campaigning on government equipment is not allowed in any branch (congress members are not allowed to e-mail campaign materials from computers purchased by tax dollars or ask their congressional staffers to do so.).

The question highlights a common misconception as to what executive orders are. Executive orders are not regulations, rules or laws. Executive orders are a performance of the office as the chief law enforcer. With that in mind, what Executive orders are issued is to concentrate resources on enforcement existing laws or to remove resources from enforcing existing laws.

An executive order cannot be issued in connection to laws that do not exist. For example, your question implies that a President can send out law enforcement to arrest people who violate some non-existent law that the President thinks up. Once the police arrest you and get you to go to court, there has to be a law that the arrestee violated. If the law does not exist, the judge will just sit there unable to inflict punishment.

So the answer is No because there is no path for the President to make any law with punishment.

That being said, there are bureaucracies that do exist that can make their own rules, exist outside presidential and congressional support and have teeth given to them by congress. That is the executive branch bureaucracies such as the EPA, FCC, USDA, etc. They can certainly eat away at the first amendment and have. If you worry about losing first amendment rights, that is where abuses are rampant.