Well, the last I checked, you just buy a NR tag across the counter. There may be some restrictions in a few units, but none I know of. Alaska has 26 units and those units near roads/population centers may have a quota of some kind, but I wouldn't hunt there anyhow. My best moose hunt was a float trip down one of Alaska's remote, fly-in rivers. Best trip I ever had. Almost all of the hunting in Alaska is a big deal. You don't just jump in your pickup and head for the hills. There are few roads, other than main highways, so you need a way to get to the game. Flying is the most popular, but some folks use boats and others use ATV s. There are lots of ways to hunt moose, but having done most of them, I recommend the float trip option. If you are planning to hire a guide or outfitter, they can fill you in. If it will be a do it yourself, PM me and I'll help a bit.
I found the best meat processor to be Indian Valley Meats, in Indian Valley, just south of Anchorage about 30 minutes. For shipping your meat home, I'd call them and see if they have an option, after they cut and freeze it. In any case, it won't be cheap. You can also check with Alaska Airlines and get prices from them. Like I said when I started this, hunting in Alaska is a big deal! Good Luck!