Green’s Dictionary of Slang

(orig. US Und.) someone bearing a sub-machine gun.

1943		E. Pyle Here Is Your War (1945) 30: He spied two tommy-gunners walking along the street. He rushed out and asked them if they would guard the ammunition.
1949		Sat. Eve. Post 9 July 88/4: A man with a sword across his knees sat on the hood of the jeep, shouting orders to a squad of tommy-gunners [DA].
1990		(con. early 1950s) J. Ellroy L.A. Confidential 456: Tommygunners charged car 3 – full clips took down the door.