There's no real way to do this besides asking the data center operators. But you seem to be going about things the wrong way. Most people look for good shared hosting providers and then pick one that has data centers in their desired region. Data centers are pretty much commodity services. It's the managed hosting services built on top of them that differentiate different web hosts. So it makes no sense to pick a data center out and then try to find which hosts use it.
Even if you're doing this because you're targeting users who live in a very small area, there's no guarantee that the closest data center will provide the best performance. Internet traffic takes the most direct route based on network topology, which isn't always the most direct or shortest route geographically. Peering arrangements, routing policies, physical infrastructure, traffic patterns, etc. all affect the network topology. So it doesn't really make sense to choose hosts based on street address.
You'd be much better off googling for shared hosting providers in Chicago and then testing the performance of sites hosted by each web host.