I am also in NY. The religious exemption is alive and well here. But here, for school purposes, you have to write a short essay explaining, in your own words, what your religious objection is. Simple clear writing is sufficient, they probably just want to be sure the objection is truly religious, not scientific or from some other basis. It does not have to be a "mainstream" religion -- you can still follow Jesus Christ's advice that you do your religious activities in private ("in your closet") rather than affiliating with a big 501c3 "religious" organizations. It's all about your personal beliefs.

As for what they do at birth, probably nothing is required. My younger child was born in another state and the pediatrician strongly recommended that mom get some antibiotic in case there was in infection in the birth canal; I believe this is standard. But the baby got NOTHING and still has not, and has been in school in NY for years now.

Healthy too.