Let me change your question slightly, and pose it to you.
Should a member of the military be able to question an order to be injected with the Dengue Fever vaccine?
Washington ordered small pox vaccinations..not technically a vaccine. It killed many soldiers.
And now we are The USA
Throughout history their have been vaccines that were problematic or turned out to not work as advertised
Soldiers will lose this case if they try to claim they cant be forced to take the covid vaccine. Will courts allow an exception based on a protected class? Perhaps..but dont think so