Yes, according to Dr. Fauci:
Contrary to popular belief, none of the vaccines have formal approval from the FDA. Instead they have been given an Emergency Use Authorization (EUA).
Under an EUA, FDA may allow the use of unapproved medical products, or unapproved uses of approved medical products in an emergency to diagnose, treat, or prevent serious or life-threatening diseases or conditions when certain statutory criteria have been met, including that there are no adequate, approved, and available alternatives. . . For an EUA to be issued for a vaccine, for which there is adequate manufacturing information to ensure quality and consistency, FDA must determine that the known and potential benefits outweigh the known and potential risks of the vaccine.
Emergency Use Authorization for Vaccines Explained | FDA
Because the vaccines are not approved by the FDA, the military cannot require personnel to get vaccinated.
Should private employers be able to require vaccinations?