I know having weapons is a big issue. Upgrading the door gives you time to get guards into the room. I have 32 residents and only 18 (might have miscounted) have guns. The way I have found them is to send out an adventurer. The in game docs say that a dweller with a high level and high special has the best chance of survival. So pick your highest leveled char also with the highest special. I'd say the special rating is your priority though. Give him the best or the only weapon. I don't think outfits reduce damage to an adventurer so only give them an outfit to boost their special unless you really need it for the dwellers assigned to your resources.
After you pick who will go, give them about 4 stim packs and 2 radaway. Send him out for one hour and watch to see if his health has dropped to half or not. You can recall the adventurer after an hour if needed but, if the adventurer hasn't found a weapon and your health has not gone down that much leave the adventurer out there longer. I had a level 10ish now lvl 15 char with high special. I sent him out and I just let him go as long as he still had health. I noticed the adventurer seemed to refill health if it levels while adventuring. On my first adventure I had left him out for 4h 30 mins and he had found one outfit and four weapons. It takes half the time to come back so you might want to recall the adventurer after one weapon just to get started. Then send the adventurer out again. I did find an outfit today after ten minutes so it's probably pretty random.