I went through this exercise a couple of times when I was of working age. If, as it appears, you want the shotgun for personal protection while you are in Kuwait, you will have to have permission and may have to ask Kuwaiti authorities. However, since you should buy one only from a street-legal gun shop, the owners of the shop might be able to give you helpful information. It is not in their interest to sell to someone who is not authorized to buy or own. The HR or security office of your employer might be able to offer assistance.
If you will be in Kuwait City, you might find it much more modern, and the shopping much more sophisticated, than you expect.
Before you buy, also ask about how and when you can dispose of the shotgun when you depart Kuwait. Leaving it in the closet is not an option. Once, on a different continent, I went through the paperwork of buying a shotgun for self-defense. Even though it was a US-made Mossberg, I could not take it out of the country when I left, nor was I allowed to sell it. I gave it back, free, to the gun store from which I bought it and did not regret that - just a cost of doing business.
If you buy a firearm anywhere overseas and want to bring it back to the U.S., go to the BATFE or ICE website or just google for the proper US government web page. Even in the 70's it was a felony to import any foreign-made, foreign-acquired gun without declaring it properly. Today, it might be a felony to import even an American-made one without declaring it or shipping it in the prescribed manner, perhaps to an FFL dealer.
This is a lot different than shipping an already-owned-in-US hunting rifles to Africa and back again, though paperwork is necessary for that, too.
Finally, ask your Kuwait employer/sponsor if they have a policy about your owning a gun in the country. You do not want to get into any kind of difficulty and then find your employer unwilling to stand by you and you really don't want to buy one if it violates company rules.
Is there a reason you think you might need a shotgun? Violent crime against foreigners in that region is, or at least was, very rare except for terrorists and a shotgun isn't likely to help you a whole lot if terrorists target you and unless you're a retired SEAL or seriously-experienced combat veteran.