The funny thing about this... there is no mainstream religion that actually bans vaccinations. Religious dogma predates the germ theory and therefore couldn't have possibly included vaccinations as anything banned.
In fact, it's the exact opposite. Most religions (at least Abrahamic religions) dictate that personal health is a paramount concern. Even if something required for good health would violate some religious law, good health overrides the religious law. For example, Judaism and Islam declare pigs as unclean animals. They are not to be consumed. However, if a pork derivative is used in a vaccine, the rule of good health means that not getting the vaccine would actually be violating religious law.
The "religious exception" was added in there so idiotic anti-vaxxers could deny their children necessary vaccinations without ever getting questioned, because asking a person about their religion is considered discriminatory.