In reality APO is an Export, like it or not, that's what the APO are and FPO addresses can be. Most deploying units have a stateside address for regular mail service and an AP/FPO when deployed.
If you don't believe me go to your local Post Office and ask. If you send to an APO address it gets a customs sticker.
In any case, the object gets in to the military Postal System and you lose track. It may or may not get airmailed, it may go in a shipping container, it probably gets handled by undertrained apes, who believe that "fragile" means "step here". Too many companies have been screwed by Soldiers and Sailor that have claimed non-delivery and the delivery time is a big unknown, so many companies don't want to have the hassle.
Companies that sell arms or military supplies may have to comply with ITAR, if they only sell web gear and non-weapons stuff, it shouldn't be a problem, IF they also sell wepons and weapons related material ITAR becomes a player and they probably don't want to mess with it. Not necessarily what I agree with but I understand having dealt with ITAR and corporate shipping, when I tried to send galvanized bolts and metal plates to Canada. (Yes I fed-ex'd them myself and charged to the company account. Could have been fired if caught) ITAR applies to weapons, communications stuff, computer stuff, riciulously complicated, most companies would raather not lose their licenses and gov't contracts over a minor violation.
So the best bet is to have it sent to a family member or friend that will re-mail it. And stay away from anything that might be ITAR listed, I think optics, or electo-optics are. Web gear, knives, etc shouldn't be a problem.