Thank you for posting your question on History Hub!
Unfortunately a search in the National Archives Catalog did not render any results for maternity homes during this time period in Illinois. What I would recommend doing is to use either Ancestry or FamilySearch to search in the city directories that they have available. Ancestry is a subscription-based database, but it is available for free public use at all National Archives facilities and many public libraries. FamilySearch is free but you have to create an account with them.
Another suggestion would be to review the FamilySearch Research wiki for Illinois to see if any additional information can be found:
Another recommendation would be to contact the Illinois State Archives to inquire if they have additional information:
We hope this information is helpful and best of luck with your research!