If I recall Illinois law correctly, a non-resident may purchase ammo there with no requirement for an ammo permit. The permit requirement is for IL residents only.
Already in existence. When my son lived in Illinois he had to have an issued buy the state permit in order to buy a box of .22 shells. We were in Walmart with him. I wanted to buy some 22 ammo. Unbeknownst to me the guy asks for my ammo purchase permit and I laughed thinking he was kidding. NOT. My son had one so he got it out and bought the ammo himself, followed by handing it over to me in the car!!