This is governed by Chicago's Municipal code in the resident Landlord Tenant Ordinance. Under the RLTO, screens are required on all opening windows. The code does not state what time of year they are required, just that they are.
City Code section 34 (13-196-560) From April 15 to November 15
- States essentially all openings from a family unit that goes directly to the outdoors. There are variations and exceptions in the full text though.