Are there reasons to not get a vaccine?

Yes. It depends on the specific vaccine. The only reason common to all vaccines, is if you are allergic to the components of that vaccine.
Yes but very few. There are some reasons depending on the vaccine, called "absolute contraindications." these are very few and your doctor should know if you have a medical contraindication, that is, a reason to not give a vaccine. Philosophical and religious reasons should not be confused with the very rare contraindications.