Best done is Free Roam. All guns will become available early in the game after unlocking the ammunition shop. If you want to do this earlier, simply raise your threat level, take out a few police men and pick up their shotgun. While in Free Roam, select the shotgun and search for large crowds, usually found near the beaches or parks.
Alternatively, you can raise your threat level so the police starts chasing you. Wait until you have 4 or more police men behind you, turn around and shoot at them.
Also during the Death Cam VHS story mission, a few enemies will drop shotguns after you kill them. Save your ammo. There will be a room in which 7+ unarmed enemies that will come out in a straight line. Run around the room until they are all cluttered into a small group and shoot them with the shotgun.