As far as it not being a religious issue,. if the Vilna Goan & the Bal Shem Tov both testified that a piece of meat was kosher and Zev said that that piece was HAM. The Talmud states that for everyone the meat is kosher but for Zev the meat is HAM. So even though for everyone else it is permissible to vaccinate but for those who hold it’s prohibited it is the same as making them eat pork.
That is totally incorrect.
You’re misrepresenting the gemara of שויא אנפשיה. It doesn’t apply where it’s blatantly and obviously wrong, so although if someone said a piece of meat was not slaughtered, or that a woman was married, the piece of meat would be assur and he wouldn’t be allowed to marry her. However, if he said it was ham, that would be blatantly false and he would still be allowed to eat the meat.
That’s probably true here as well.
Antivaxxer meshugas isn’t based on the antivaxxers claiming to have witnessed something, as the cases in the gemara are, it’s just that they’ve accepted the false narrative of a bunch of quacks instead of mainstream medicine.
Any posek would tell an antivaxxer that it’s a chiyuv to vaccinate, and that their meshugene dayos don’t change that.