Green’s Dictionary of Slang

1. easily defeated, complaisant.

1920		A. Baer ‘Bugs’ Baer Says 1 Oct. [synd. col.] ‘The gambler goes around and fixes up a round-heeled series.’ ‘What’s that?’ ‘A pushover’ .
1950		Goldin et al. DAUL 181/1: Round-heeled. To be easily victimized or used.
1962		B. Veeck Veeck — as in Wreck 140: Frank, of course, is round-heeled when it comes to a trade. [...] I mean that if you come to Frank with a proposition, he can’t say no.

2. of a woman or gay man, promiscuous.

1957		J.D. MacDonald Price of Murder (1978) 73: Silly bitch. Lee deserved the best, not a round-heeled bag like that.
1959		W. Burroughs Naked Lunch (1968) 122: Round-heeled boy, one greased brain goose and he flop in an obscene position.
1989		H. Rawson Dict. of Invective (1991) 331: Round-heeler and round-heeled also have been sighted.
2010		J. Ellroy Hilliker Curse 14: Fleet #2 tailed the roundheeled redhead to juke joints and hot-sheet motels.