How do we obtain military clearance (if we reside on a military installation) to become foster parents?

You will have to notify the Child and Family Services off base. They are the ones that will start the process for adoption and foster services.
JAG, your husband’s unit and housing don’t have nothing to do with getting things started.

Once you get in contact with CFS, they will let you know what you need to do. I can tell you that both you and husband will need:
-background checks
-fingerprints done
-Drug Tests
-Home Inspections
-Health Screenings and a Check-up
-interviews with you and hubby and if you have any children in the house depending on how old they are, your children will also be interviewed.
-You will have to attend adoption/fostering classes through Child and Family Services (or somewhere else if they don’t do the classes but they will let you know).

Once you have started the process and things get going, then notify housing about what is going on. You will then have to let them know when you get a foster child in the house. You would be surprised to know that military families on post are in high demand because they try to keep military children with other military families for some stability especially if that child lives on base on and goes to on-post schools.

If you want to feel a little better, you can contact Housing the same time you contact Department of Child and Family Services because posts do differ a little bit.

We are currently going through this process however to adopt not foster.

Good luck.