I believe that the decision to give children vaccines should be between the parents and doctors. I believe in informed consent for any medical procedure.
Currently, the vaccine mandate includes any school in California, including private schools. The state government appears to have the right to mandate vaccines because they run the schools, based on supreme court precedent. However, mandating that private schools require vaccines is clearly wrong, although the argument for it is "public health".
Parents should definitely have the right to choose a private school that does not require vaccines if they and the school so choose.
Currently, the only way to avoid vaccination is to home school or get a vaccine waiver, which in California is nearly impossible.