Unless your state has unique laws governing this in private schools, I think they generally have the legal right to deny you admission to the school if you don't meet their requirements (be that vaccination or test score or interview placement or parish member status). Most states even have laws allowing exempting religious institutions from the laws barring discrimination on the basis or creed or religion. I would try to find a different school you liked as well where this would not be an issue. While some Catholics opt not to vaccinate on a personal level because of some vaccines being developed (long ago) from cells from aborted fetuses, the church formally does not have an opposition to vaccinations. Catholic schools do enroll those that are not Catholic but they don't have to do so. It's my understanding that many Catholic schools don't accept exemptions for vaccinations unless it is medical.
You could try to talk to his Bishop and see if they will override him. That is all I can think of besides finding a different school.