The word chick is a short form for chicken and its use in American slang to refer to a young woman is attributed to Sinclair Lewis' book Elmer Gantry (1927).
He had determined that marriage now would cramp his advancement in the church and that, anyway, he didn't want to marry this brainless little fluffy chick, who would be of no help in impressing rich parishioners. But that caution he had utterly forgotten in emotion, and her question was authentically a surprise, abominably a shock.
c.1320, abbreviation of chicken (q.v.), extended to human offspring (often in alliterative pairing chick and child) and used as a term of endearment. As slang for "young woman" it is first recorded 1927 (in "Elmer Gantry"), supposedly from U.S. black slang, in British use by c.1940, popularized by Beatniks late 1950s. Chicken in this sense is from 1711. Sometimes c.1600-1900 chicken was taken as a plural, chick as a singular (cf. child/children) for the domestic fowl.