The order has already been given: all members of the military on active duty or ready reserve must be vaccinated against COVID-19.
https://media.defense.gov/2021/Aug/...-OF-DEPARTMENT-OF-DEFENSE-SERVICE-MEMBERS.PDF
It proceeds under the same guidelines as the existing military vaccine program - meaning that servicemembers who have "contraindications" (i.e. allergic reactions) are exempt, and that all existing administrative exceptions (which includes
religious exemption) apply.
It should be noted that the process for obtaining a religious exemption from vaccination includes counseling from both doctors and command on the medical and career consequences of not being vaccinated. It seems that this counseling is what the SEALs are objecting to, calling it discrimination. Assuming that they do obtain a religious exemption, I see no indication that they will be court-martialed or discharged for not being vaccinated. However, they may be removed from Special Operator status - which I feel is reasonable. COVID is a threat to the readiness of our armed forces. Should our most elite soldiers be placed on active duty lacking full protection against it?
I will say that seeking the recovery of their training costs might be excessively punitive, but that's the only thing that stood out to me.
ETA: there may also be some legitimacy to the complaint that those who get medical exemptions will keep Special Operator status/pay while those who get religious exemptions won't, and it seems like some individual commanders may have taken things too far in trying to encourage their troops to get vaccinated. But I see no issue with the order itself.