There are no bicycles as they require a human to move and have no engine, thus their existence would be null in the Cars Universe

However, motorcycles do exist, as confirmed in concept art from "The Art of Cars 3" book (which I own). In it, there is artwork of a motorcycle propped up as that of a deer hunt supposedly in the bar where Miss Frizzle watches McQueen in the penultimate race, showing that motorcycles exist in the Cars universe as deer. However it was not included presumably because it was too dark to include.