That's actually the DoD position. Once the vaccines are given full authorization, they will be required by DoD. But until then, the DoD (and DoS) rule is that only fully vaccinated personnel can go without masks.
However, the rule up to now has been a sort of "don't ask, don't tell" approach to people at work in government facilities -- at least on the State side. Vaccinated employees could drop the masks, but we couldn't ask anyone whether they had been vaccinated. I had to deny entry to this country to military members who were not vaccinated, but had to approach it from another angle: our facilitation of DoD who wanted to come here was an exception, and would only be made for vaccinated personnel. I got some push-back ("it isn't our policy to pull someone from a team," or "we can't punish anyone who chooses not to take the vaccine"), but I countered with the fact that not being permitted to engage with a foreign force was not a punishment. But refusing vaccination wasn't without consequences (not punishment). Luckily for my office, PNG is now only allowing vaccinated travelers to enter the country, so it's out of my hands!
Looks like this Alabama base is having an integrity issue. Again, they're not saying everyone has to be vaccinated, but they are enforcing the "only vaccinated personnel can unmask" rule with more oversight.