Not that hard of a question, I think. A company should NEVER have any say in what I do with my health, other then what is done by the law.
They should not even KNOW if I take vaccinations or not.
If the general public decides that not having a vaccination should be a reason to not give you a job, then put that into the law. The companies then follow that law.
I once had a cow orker who was told he could only keep working if he would take antabuse. They are NOT a doctor. They can NOT prescribe medication. One day he was fired because they said he did not take his medicine. He had a field day, because he was able to take them for everything he was able to.
So no, a company should NOT be able to decide on medical issues, except those provided by the law. And those are mainly directed towards the safety of the individual, not of the company. e.g. sleeping time for long haul drivers. Maximum hours work. Minimal hours of breaks. Not working when it is too cold or too hot. Providing free drinks at certain temperatures.
That said, they might encourage you to get e.g. flu shots. Where I work they will pay the flu shot for you if you want, but nobody can be fired if you don't. And never would they DARE to oblige you to go. That would be such a huge invasion in the privacy, that they know they would loose any lawsuit no matter how badly prepared you were and how much money they have. It would be the shortest trial ever.
Judge:"What? Guilty! Next"