Pennsylvania law regarding preschool vaccination requirements and exemptions applies exactly the same to public and private schools.
Wait until your child is accepted into the school before revealing that she will be using an exemption. If you let them know before she is accepted, they might not let her in and say it is for a different reason.
If they don't follow the law by allowing the exemption, surely they would be willing to give you your deposit back.
Pennsylvania vaccine exemption law for childcare and preschool only mentions a religious exemption. The law applying to kindergarten and higher grades also mentions "strong moral or ethical conviction similar to a religious belief." So be sure to keep your exemption letter purely religious for preschool.