There was a good discussion of that topic here, of all places:
https://slashdot.org/comments.pl?sid=19410631&cid=61625009
Basically, it seems to have a very minor impact. There is existing precedent for being forced to vaccinate, though I prefer it be used as a very last resort because I think that being forced to do a thing makes more people oppose it and I'd rather see more light and less heat in the vaccine discussions.
There are real risks to vaccines, but according to VAERS data, they're orders of magnitude smaller than those of the virus itself. If one isn't concerned by a 2% mortality from Covid, it seems absurd to me to worry about a ~0.0002% chance of vaccine injury.
If you were playing an RPG and got a magic item that let you roll 1d1000 every time you died and escape death on every roll but a natural 1, you'd think that was OP. This is effectively what the vaccine is, so it makes no sense to avoid vaccination.