Records of orphanages can be difficult to locate. As many were private institutions, their records may not have survived. Local genealogical and historical societies may have some of those records. If you know the general location of the orphanage, there is a listing of those in Illinois at http://www.formerchildrenshomes.org.uk/illinois.html. On the Census, the orphans may have been listed as “orphans” or “inmates” which helps locate these homes.
If your ancestor was born after the 1880’s, when most counties were recording births, AND they were adopted, it may be possible to obtain their original birth certificate through the Illinois Department of Public Health at https://www.dph.illinois.gov/topics-services/birth-death-other-records/adoption
If your ancestor came to Illinois on an Orphan Train, there are a few additional resources to check. Between 1854 and 1919, over 100,000 children came from New York city and were sent west for adoption and hopefully, a better life. The National Orphan Train Depot Museum has a lot of information on these children. Their website is https://orphantraindepot.org . ISGS also has reprints of the book “Children of Orphan Trains from NY to IL and Beyond” for sale on website at www.ilgensoc.org.
Note: This was originally published in the "Tips from the Genealogy Committee" column in the Illinois State Genealogical Society Newsletter (May 2021). While these tips were written for those researching Illinois ancestors, many of the principles can be applied to other locations.