Is the public entity the party that you are suing? If so, then a subpoena is likely not the right way to go to get the documents you seek. Instead, you would do that via discovery. Just because the lawyer on the opposing side was on the council would not make him/her the party that holds the records. Indeed, it would be unusual for that to be the case. Rather, the holder of the records here is likely the public entity itself. And if that is the party that you are suing, then discovery is likely the proper way to get what you are looking for.

In any event, assuming that these records are not protected from disclosure in civil litigation under IL law (and I have no idea if they are as I know nothing about lawsuit or the public entity involved) then generally what you would do is serve the discovery request or subpoena request (as appropriate) and if the opposing party/party being served objects to the release of the records or claims that there is nothing relevant in them, then depending on the nature of the dispute one side or the other may ask the court to undertake an in camera review of the documents. That is a review of the records by the judge outside of the presence of the parties in which the judge can look to see if there is anything relevant in the documents, whether any privilege or immunity applies to them, or whatever the dispute is about.