When I was in practice I had a few parents who refused vaccines despite my careful explanation of the benefits and risks. One that I distinctly remember was an Hassidic Jewish mother who claimed, although there was no basis for vaccine refusal in their doctrine, that a rabbi had advised her to follow her conscience on this. I thought this odd, but at least she claimed to have pursued some discernment for what she claimed had felt was God's will for her children.
There is certainly nothing in the rudimentary doctrine of Friends that supports not vaccinating our children. Medical care is a gift from our Creator, improving and saving lives. Picking and choosing which medical care to help their children seems not to be based on true spiritual discernment. I think rather my friends have gotten caught up in the Internet-enabled anti-vaccine movement. I asked them to seek true discernment and accused them of being less than truthful.
Another issue with vaccine refusal is the jeopardizing of others health. Vaccines are a public health concept. We get vaccinated to protect ourselves and others. When particularly susceptible children who do not have protection against a particular illness either because of age, underlying illness (AIDS, cancer) or lack of efficacy of their own vaccine (none are perfect) catch something from the unvaccinated, well that is just wrong. When we protect our own children we help others as well.
Although claiming a religious exemption might have legal grounds, I am not clear that it has moral grounds.
Who would Jesus vaccinate?