The Chaplain only reviews the request and forwards his thoughts to the CO. The CO endorses the request with a recommendation, either for approval or disapproval. That request is forwarded to the first Flag Officer in the chain of command, who disapproves the request.
FWIW, in 30 years, I never saw a religious exemption approved for a vaccine. Not once. And that included the years John Paul II was Pope, and he was vehemently opposed to vaccines that were tested with fetal stem cell lines. This is not a new stance for the military.