Are there any pay-out-of-pocket therapists/psychiatrists I can see in Chicago, IL?

All therapists, including psychiatrists, will accept your money! You just need to find a good one with an opening. You have to be willing to speak to people because word of mouth references are the best. Otherwise, call up the Psychology department and the Psychiatry division in the universities near you, U of Chicago for example, and ask them if anyone on their faculty sees patients outside of school.