There are only a few ways a tenant with a section 8 voucher can be turned down in Chicago.
1. an owner occupant can discriminate when choosing a tenant for the building they live in.
2. an owner can have a building that will not pass a CHA inspection
3. the voucher holder cannot meet basic creditworthiness criteria used by the owner to evaluate applicants.
4. a Voucher has restrictions on the areas it may be used in, based on neighborhood income, blighted status or other conditions not related to the specific property.
As someone who has been working with voucher and opportunity (mobility) voucher tenants, I can tell you #3 is what continually keeps the door closed to section 8 voucher holders.
Almost any other reason is a fair housing violation, but an owner has the right to require all applicants to conform to a creditworthiness standard.
Know your FICO score, and ask when you apply what the credit score criteria is for an owner to review an application, and be sure you get a specific number, not a statement like "good credit", and remember income alone cannot be used to determine whether or not an applicant can be accepted as a tenant.