You may want to check into the regs and instructions a little closer before you brew while in a deployed setting. There are a lot of places you could be going but my first guess is the Horn. I haven't reviewed the instructions there or anywhere specific to your situation, but if you run amuck of a little known instruction you could be looking at a quick ART 15 or Court-Martial. An ART 92 Orders Violation (General or otherwise) is real easy to slap on someone.
I would hate to see anyone take it on the chin, at least without knowing what the possible outcomes are.
With that said, if you can do it without getting hammered, I believe Northern Brewer ships APO. When I was in Japan, my buddy had stuff shipped to him and I think it was Northern Brewer.
Also, it may take some digging, but there is sometimes a CONUS based address associated with your APO address. We could sometimes ship stuff to a random CA address and it would forward out of there to us in Japan.
Good luck, Thanks for your service and stay safe.
"Life is hard; it's harder if you're stupid" John Wayne
"I can imagine no more rewarding a career. And any man who may be asked in this century what he did to make his life worthwhile, I think can respond with a good deal of pride and satisfaction: 'I served in the United States Navy.'" JFK