I've used RFD a couple of times. As I pointed out in post #54 of this thread, RFD does have limited international service, and would be my candidate for the airport that best answers the question posed in the title of this thread.

The RFD-CUN flights are scheduled to operate 16 times this winter season. They are marketed by Apple Vacations and flown on Allegiant Air equipment. And yes, RFD does have an immigration area. How else could they handle non-stop inbound flights from Mexico? The "immigration area" (that is, the international arrivals gate) is not located in the passenger terminal. It is in the RFD administration building, which may be why you aren't aware of it.