Archery tags are over the counter, so you can still get them. They are good throughout the state (unlike rifle tags, that are limited to a particular county or public site). If you are on the north side of Chicago, any public land within an hour and a half will can get fairly crowded. The closest sites to you are Chain of Lakes, Volo bog, and Morraine Hills, the latter 2 require you to draw a permit at the site in addition to you archery tag (details at IDNR website). There's some public land, probably more like 2+ hours to the south and west (again, see the IDNR website). I'm not sure how long it takes to establish Illinois residency, but if you can't by the time you have to buy permits, they will be very expensive. Your right that your best options may be to the north. Wisconsin has a lot more public land (not sure about an hour and a half, though) and they don't gouge nonresidents the way some states do.