If the boat will reside in Illinois, and you're going to register it in Illinois, then you have to register and title the trailer in IL as well. The only way you could get away with it is if the boat and trailer are registered in WI at a WI address, then the trailer doesn't need a plate, and doesn't need to be registered.
Just researched all this because I had debated just not bothering with Illinois, and using a friend's address in WI to register the boat and trailer I bought over the summer. Ended up just registering both in IL, and transferring title on the trailer and getting a plate.
From what I remember, if you have no title on the trailer, you have to apply for a "homemade trailer" title, I believe, from the Illinois DMV.