No need to worry. Public schools cannot even give children a tylenol without parental/physician approval. If the vaccinations are offered at schools, each child will have to have parental permission to receive a vaccination. No permission, no shot. The laws have changed radically since I was in school and they routinely came around with those guns that gave vaccinations for every child. In Georgia, schools have to have a physician approved release for a child to get tylenol or even neosporin. My wife is not even allowed to give lotion to a student.
On the whole, that’s correct. They can’t even give tylenol to a kid without parental permission. But they can give condoms to minors in many school systems. And in many states, they can take a kid to a clinic for an abortion without notifying the parents.
So, I suspect that if the government really wants to do this, they will find a way to do it, one way or another.