If you entered the U.S. Armed Forces from Illinois and are an Illinois resident, you can hunt and fish without a license while you are on leave. You still have to buy a Federal Waterfowl Stamp and an Illinois Habitat Stamp to hunt.
There is nothing to apply for. Carry proof of your Illinois residency and your leave paperwork while you hunt or fish, and buy the required stamps.
Proof of Illinois residency, your leave paperwork or military ID, and your Federal Waterfowl Stamp and Illinois Habitat Stamp for hunting.