Is it possible for a mother of a military dependent to get a military id if she is not in the military?

It depends on the situation. Generally speaking, not unless she can no longer take care of herself. A common misconception is that if a military member supports someone, that person becomes a dependent. This is not true. The mother would have to be declared legally unable to care for herself and then the child would have to support her and file the correct paperwork. Even then, though, her ID would only entitle her to use of the exchange, commissary, fitness center, etc. It would not entitle her to TriCare. She could use the MTF (Military Treatment Facility) on a space-available basis, but MediCare and her insurance company would have to cover the costs.