Federal law requires that you be a must be 21 years of age and a resident of the state in which the firearm(s) are to be purchased. There is no exceptions for active duty military personnel. I'm retired military, live in Texas and know for a fact you must be a resident. If you decide to get a Texas drivers license, you might want to also think about changing your official residence to Texas with the military. You will not have to pay CA income tax and TX has no state income tax. One other thing to consider, be careful of what you buy if you eventually decide to return to CA. Generally, most firearms are likely not legal in CA, so look for CA-specific approved firearms.

Considering how California gun laws are structured, you are going to have a hard time getting anything transferred to you. If you aren't going anywhere in the near future I would recommend just going and getting a Texas DL and avoiding a massive hassle.