Green’s Dictionary of Slang

(US) a vagrant, a tramp.

1895		L.A. Herald 15 Dec. 21/1: The fakirs [...] haunt the country districts, and the funny papers called them ‘Wandering Willie,’ ‘Dusty Rhodes,’ ‘Weary Waggles’ and ‘Rusty Raggles’.
1900		Riverina Recorder (Moulamein, NSW) 5 Dec. 2/3: Wandering Willie has not hooked his fish yet.
1916		‘A-No. 1’ Snare of the Road 30: A continual repeating of the self-same theme innoculated the Wandering Willies [...] with gambling spirit.
1933		‘Goat’ Laven Rough Stuff 50: The sight that greeted us as we stepped into the station was about two hundred tramps and bums and wandering willies, all cuddled up on the benches.