Congress may generally condition direct federal assistance on any basis it chooses, provided that no suspect classification is used, and that the law passes rational basis scrutiny. That is, congress must assert, or the courts must find, some plausible connection between the restriction and the purposes of the act.
Funds distributed through the states, and particularly ones partly funded by the states, are subject to different and stricter constraints. Congress does not have as free a hand in those cases.
In this case, it is valid a purpose of welfare legislation to encourage beneficiaries to find and accept paying jobs when possible and as soon as possible. Congress might think that being vaccinated makes that more likely.
However, no such federal law has been passed, and politically I think it is unlikely that one would be. Until and unless such a law is passed and then challenged, one cannot be sure if it would be found constitutional.