Wasn't the de facto Republican candidate for Mayor of Chicago Paul Vallas in 2019?
Vallas is a Democrat and served as Pat Quinn's LG, so that shows how far Republicans are from actually getting one of their own in the game.
Vallas was endorsed by Bruce Rauner and the Chicago Republican Party, so there was some GOP support for him. Other Republican support tended to go to either Willie Wilson, Garry McCarthy, or maybe Bill Daley. Together, those candidates combined for about one third of the overall vote in the first round (though of course not all of their supporters are Republicans).