Both Title VII of Title VII of the Civil Rights Act of 1964 (as amended), and the Americans with Disabilities Act(ADA) as amended, provide that employer mandates are subject to "reasonable accommodations" for "sincere religious beliefs". If any mandate was based on or subject to either of those laws, requests for accommodation would need to be addressed on a case-by-case basis. The belief does not have to be a tenant of any church or organized religious group, but may be purely individual.
State laws granting religious exemptions might also apply.
However, in Prince v. Massachusetts, 321 U.S. 158 (1944) the US Supreme court wrote:
Thus, he [a parent] cannot claim freedom from compulsory vaccination for the child more than for himself on religious grounds. The right to practice religion freely does not include liberty to expose the community or the child to communicable disease or the latter to ill health or death. People v. Pierson, 176 N.Y. 201, 68 N.E. 243.
Prince was a case of a child labor law (selling religious pamphlets in the streets) not an actual vaccination case, so the above statement was not strictly binding precedent. Whether it would now be considered good law I cannot say unless it coems up oin a current case.
To the best of my knowledge, no major religion or denomination objects to vaccines as such, although some do object to vaccines developed using fetal stem cells. I believe that at one time the Jehovah's Witnesses did so object (one of them was the appellant in Prince) but they no longer hold that view. At least some individuals have expressed such objections, but most expressed objections to vaccines or vaccine mandates have not been on religious grounds.