Green’s Dictionary of Slang

(US drugs) an injection of narcotics directly into a vein.

1936		D. Maurer ‘Argot of the Und. Narcotic Addict’ Pt 1 in AS XI:2 127/1: vein shot. An injection of narcotics directly into the blood stream.
1959		J.E. Schmidt Narcotics Lingo and Lore.