From Wikipedia's entry on "Locked Room Mysteries".

I'm guessing: suicide, made to look like murder for the insurance. The gun was on a string threaded through the transom, and an accomplice retrieved it.

Alternatively, the Laundry owner was one of three brothers who had previously escaped from a Russian prison by hiding in coffins. The "body" was alive but shamming and made-up with fake blood and burns marks; the real body, the guy's brother, was swapped in the morgue later.

Alternatively, Mulrooney was the murderer. He sent a trained dwarf assassin through the transom who shot the (dead drunk) victim with a silencer before secreting the gun back in the transom and then opening the door.

Alternatively, the whole of Fifth Avenue was mocked-up on a film-set in Burbank, and dozens of New Yorkers drugged and transported there before being drugged and flown home again.

Alternatively, aliens.

Alternatively, the murderer hid in one of the big drums of the Laundry's equipment until everyone had gone.

Alternatively, the gun was fixed up out of starched cuffs, fine soap powder served as gunpowder, buttons for bullets; Fink committed suicide and the weapon he used was bundled away with the rest of the laundry.