I do not see where the OP mentions Base housing, just that the OP lives in GA, in Government housing (subsidized apartments one possibility, as is a Military base).
Since a single animal can be cared for by more than one individual it would depend a lot on what the therapist(s) feel was the necessary ingredient in having the ESA and if that singular item is possible to be a shared responsibility. Things that are needed, grooming (brushing/combing animal), feeding, exercising (playing with the pet indoors or outside), walking a dog for toileting. All these items can be shared, and if needed put on a schedule so that each child gets to experience all aspects.
A landlord can limit the number of pets allowed, which is essentially what an ESA is. So 1 per family unit I think is a reasonable accommodation, if one lives in a no-pet situation, unless a therapist feels the children each need their own, in which case you would need documentation specifying why more than one is needed.
I am not familiar with base procedures other than it needs to go through the base commander.