Search the web for something like "Request for Religious Accommodation Vaccine." It's related to the person's personal beliefs, so even if they're part of a denomination that doesn't have rules about it, the denomination doesn't matter. It's the personal belief. So find a passage or two in the Bible and quote that. Or some Buddhist text if you are worried about looking like a fundie. Don't mention the denomination if it isn't one of the very few that helps. So like don't say Catholic since that will just give them something to give you trouble about since the Pope wants you to get vaccinated, etc.
Don't mention anything medical, like why you have concerns about it medically. Only religious.
Keep it short, like 3 paragraphs.
They can't ask for your pastor's note. It's about your personal belief.
It's a federal law. Some companies and government branches, and California say they won't take religious exemptions but you really probably have a case since it's a federal law and it isn't up to them.