Whatever you do don’t resign if you can help it—make them fire you as it may open opportunities down the road for being made whole
Secondly, there are more than a few doctors that can legitimately look at your medical history, allergies, prior vaccine reactions if you have had any, etc and write you a letter recommending against it
If you have a letter from a doc and they are not willing to make reasonable accommodation they are probably exposing themselves under ADA
Good advise. Although, I just got another job offer today. I am a recently retired Navy Jet Pilot (Commander, ret) and I know I can get other jobs if I need to.