As far as I'm aware, you can't buy a gun from an FFL (a gun store, or anyone whose business involves selling guns) in a neighboring state without presenting an Illinois Firearms Owner Identification card. And most (all?) gun stores will refuse to sell you a gun that they know would be illegal in Illinois if you live there.

Most of the guns used in crimes in Chicago are illegally owned handguns, acquired through straw purchases and theft. Handguns that are legal to own in Illinois, even.