The Illinois Department of Natural Resources website (dnr.illinois.gov) is your best source of information. I believe you should be able to buy tags over the counter once they are available. Please check to be sure as I am unfamiliar with non-resident permits. At a minimum, you will need a hunting license, a habitat stamp, and a deer permit (either sex or doe/fawn only). All the above can be purchased on the website. Hunter Education course may apply depending upon your birth date.
Jim Edgar in west central IL. is a very large tract of land in great whitetail country. Shawnee National Forest in southern, IL. is even bigger. Both offer free public access. Jim Edgar will likely have a few more rules than Shawnee. For example, designated parking areas. You will need to cart or drag your deer out of there as they do not allow vehicular traffic.