Godspeed Navy anon.
As a young AF anon I didn't like vaccines because they usually made me sick. BUT, I knew i didn't have a choice. So I tolerated it for 4 years, and it was part of the reason anon decided not to re-enlist.
Seems to me as a MIL member, they DO have the power to force you to vax. I don't think resisting would stand up to any kind of legal review. You are brave for being willing to force a Court Martial, but I think they will just give you separation papers if you won't comply with the vaccine dictate.
THEY can terminate YOUR contract at anytime, just not the other way around.