Most, if not all, military installations will have a post office on them regardless if they are stateside or overseas. They should be able to handle all of your shipping needs.
The only concern I would have would be finding the cardboard boxes for the 2nd layer of protection, i.e. 9x9x9 boxes. If you have an idea of how many balls you plan on purchasing, I would suggest you bring the cardboard boxes with you and a roll or two of packing tape to ensure you have it. For example if you plan on purchasing 4 balls, talk with your local pro shop about getting an empty case from them and using that to mail your purchases back.
Another suggestion is to use the Japanese postal system or maybe there is a UPS or FedEx or DHL office in the local area (of the base) that you can use.
Bottom line...there should be no reason for you to not be able to ship some bowling balls back.