In Illinois, the "official" way to find the "official" address of a business appears to depend how the business was organized. That is, the business could just be someone operating under an assumed name, or it could be a corporation, or it could be a limited liability company, etc.
For an assumed name, the name and address must be registered with the county. See (805 ILCS 405/) Assumed Business Name Act.
A corporation must have a registered office in the state. See (805 ILCS 5/) Business Corporation Act of 1983.
The laws pertaining to other business organizations appear similar, and it looks like you can find a whole list of them in the Illinois Compiled Statutes. The link lists the laws pertaining to the various business organizations in about as clean a format as I have seen.
Now, it appears that the Illinois Secretary of State has a page allowing you to search for information about a corporation or LLC. However, since assumed names are just registered with the county in Illinois, you would have to find the county administration offices to get that information.
Finally, if the business is in violation of the registered address laws, you may have to discuss with a (perhaps different) lawyer how to serve them. In Minnesota, it is permissible to serve process on the Secretary of State for certain similar circumstances, but I do not know whether such a mechanism exists in Illinois.