One that slips into houses in the dark, to steal cloaks or other clothes. Also lambs’ fur formerly used for doctors’ robes, whence they were called budge doctors. Standing budge; a thief’s scout or spy.
Definition taken from The 1811 Dictionary of the Vulgar Tongue, originally by Francis Grose.
See also the definition in Nathan Bailey's 1736 dictionary of canting and thieving slang.
Buckles * To BudgeNathan Bailey's 1736 Dictionary of canting and thieving slang
John S. Farmer's collection of canting songs and slang rhymes