The M1881 20ga Springfield "Trapdoor Forager" is the earliest purpose issued US shotgun I know of.
Prior to the adoption of the rifled musket and except for a few issue rifled small arms, virtually all muskets, carbines, musketoons etc. were technically shotguns. They were capable of firing buckshot, shot and ball loads. Geo Washington favored the combination buck and ball load and prescribed its use in battle, although it is doubtful this was possible on a large scale due to poor logistics. When the troops had to forage for small game with, they loaded up with a lighter charge, some wadding and shot.
One of the Marine officers killed at the Battle of Chapultepec was armed with a double barrel shotgun and there is a story that a Private in one of the volunteer units brought an early Marlin or Winchester pump shotgun to the Philippines around 1899 for use on opium fueled Moros, setting the precedent for the modern pump combat shotgun. (Today, autoloaders are issued, but in much smaller numbers than the pumpgun.)
The Winchester trenchgun was first issued in 1917 as far as I know.