A religious exemption does not have to be signed off by your shul rabbi; it has to be based on sincere personal religious beliefs. If someone feels that the risks of vaccination outweigh the benefits, then that person would have a real halachic issue putting his healthy children at risk for the theoretical benefit of the klal.
I don’t understand. If you say it’s a real halachic issue, then you should be able to produce a letter signed by a legitimate rav stating so. If you can’t produce such a letter, then any “sincere religous belief” you have is apikorsus.