Gerald is correct about St.Vincent's Orphanage, but that was mostly for babies. Once the babies were old enough, they were either adopted out or sent to another orphanage, once they were potty trained or old enough, around the age of 3.
One of the Orphanges was Angel Guardian Orphanage (A.G.O.) which I had attended.
You didn't say how old the children were at the time, so they were not babies, they were not sent to St. Vincents.
A.G.O. was also run the Catholic Charities at the time, before the State took over. You can get information by contacting the Archdiocese of Chicago.
I do have a list of names of other Orphanages in Illinois that you may want to check out, but the majority of them have closed down.
If you would like any other information that I can supply you with, you can contact me directly at email@example.com