Green’s Dictionary of Slang

to follow a person, to place someone under surveillance.

1928		J. Callahan Man’s Grim Justice 47: We’ve got to tab the ‘night hack’ (watchman) [...] get a line on his movements.
1929		D. Hammett ‘Fly Paper’ Story Omnibus (1966) 33: It was simply a matter of sending a man around [...] to pick up whatever information he could from her friends and neighbors, without, of course, letting her know she was being tabbed.
1949		Monteleone Criminal Sl. (rev. edn) 232: tab To keep in touch with.
1954		(con. 1940s) E. Lambert Veterans 131: There’s a bastard up in a tree over there [...] I think I’ve got him tabbed.