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Definitely like this idea. But there are some possible issues to at least consider.
1) We need to make sure the vaccine is perfectly safe for everyone.
If there are any conditions that make it questionable for a person to get it, it wouldn't be fair for them to not get a check because they medically were advised against a vaccine. Or they were legitimately concerned with getting it for other reasons that I'm sure will come up with some edge cases (not talking about people just against vaccines, but perhaps other reasons we aren't yet thinking about)
2) Would be difficult to keep track of all of this.
The IRS and local/city/state govts would all have to work in concert with one another to pull this off. It was hard enough for the IRS alone to work with a few other federal agencies to get everyone their first $1200 checks. So imagine how much more complicated it would be if you added this qualifier. You would have to pull off a very impressive coordination with local and federal govts that I think would be difficult and create challenges (and ways to exploitation) that we can't quite see without giving it a try.
Saying that, I like your logic. It would speed up the process of getting everyone shots by providing monetary incentives. How American is that?