If your employer is offering you a chance to submit a religious accomodation/ exemption form from the Covid shot, bear in mind :

1. They will ask, but per the 4th Amendment of the US Constitution they CANNOT FORCE you to disclose what your religion is
2. Congress can NOT make A LAW that prohibits your free exercise of your religion
3. Per Title VII of the Civil Rights Act of 1964. you cannot be fired for your religion

This vax mandate is Not A Law. You don't have to tell them anything about your religion, ONLY that your religion does not permit you to be Covid vaxxed or Covid tested. And they can't fire you for it.
They CAN say your not-being-vaxxed is a "hardship" for the company.
So THEN - MAKE THEM prove hardship