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Blue state California has a huge population, 39.5 million and urban rents are high. There are 130k homeless people. New York State has another 90k homeless people.

Red state Wyoming has only 500k people, with a 2018 voter turnout of around 200k.

If the State of California or a group of private citizens acquires cheap land in Wyoming, builds a city, fills it with homeless people, and supplies them, with the intention to capture the congressional /Senate seats, what could the Wyomingers do to stop them?

Nothing

This has been done in American history, in fact, though by religious groups rather than states! For example, see generally Nauvoo, IL, a city that the Mormons built to consolidate power, but there are other examples.

What stopped the Mormons from using this population growth to wield political power in the state of Illinois? Violence.

What you are proposing is perfectly legal. It is just logistically difficult (move these people, provide them a working city, ensure that they are inclined to vote the way you want without bribery), massively expensive (Billions, if not tens of billions of dollars), and one wonders who would foot the bill for such a scheme.

Nothing. It's a constitutional right.

The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.

Source: Privileges and Immunities Clause (U.S. Constitution, Article IV, Section 2, Clause 1)

Does this make sense?

In a comment, you said

California won't lose seats because they have surplus blue voters.

The number of "blue" voters is irrelevant. Under current law, apportionment is done by residents. If California moves some of its residents (even if homeless) to Wyoming, then apportionment will shift seats towards Wyoming and away from California. Now, if California did this with just the right number of people, they might not lose a seat and Wyoming might not gain.

Wyoming would need to add six or seven hundred thousand people to gain a seat. Right now, the only reason they get a seat at all is that every state is guaranteed one. They actually have fewer people than half of Montana. Without the guaranteed seat, Montana would have gotten two before they got one. Another state for comparison is Rhode Island, which just barely has two seats.

The next issue is that homeless people don't generally vote. Since Wyoming has a large surplus of conservative voters, a shift of few enough residents to avoid California losing a seat might not change Wyoming's voting habits.

What could Wyoming do?

Wyoming could reinforce this by requiring people to have mailing addresses and/or photo identification to vote. Since homeless people have neither of those things generally, they won't be able to vote. Currently Wyoming does not actually require this, but they do require that prospective voters show some form of identification. Wyoming voter registration requirements:

• 
  

  Present a valid Wyoming driver's license (If you do not have your current valid driver's license with you, you must provide the number along with additional acceptable identification as defined in Rules Ch 2: Identification for Election Purposes);

  
  
  
  • If you have not been issued a Wyoming driver's license or yours is expired, you must provide:
  
  
  • A statement to that effect;
  
  
  • The last four digits of your social security number; and
  
  
  • Additional acceptable identification as defined in rules.
  
  



• If you have neither a valid Wyoming driver's license nor a social security number, please indicate this by checking the appropriate box on the Voter Registration Application form and contact your County Clerk.

So it is apparently up to the county clerk to verify the eligibility of people without a driver's license or social security number. Voters also have to not have had their voting rights suspended due to a felony conviction and must not be mentally incompetent. Many homeless would fail one or the other of those tests.

This already happens

All that said, New York City is actually doing this. They send homeless people to live elsewhere and pay the rent for a year. But they aren't doing it to colonize or get more seats. They have just found that it is cheaper to support someone almost anywhere else other than New York City. Their goal is to reduce their homeless population.

Other cities have similar programs, but they generally focus on moving homeless people closer to their support systems, e.g. family members. New York City's program may be unique in that it just tries to move homeless people away from New York City. The same people may become homeless again after a year. It's unclear if or how many return to New York City.

The reason this does not currently happen is that it would cost the state of California to move all those people to Wyoming. one for building a new city and 2 for buying all of the lands for the homeless people. most likely the state taxes would shoot up and everybody would move and the state of California would lose a lot of congressional seats making it an unwise decision.