According to the Morel Sigthings Map, last year, fresh morel mushrooms started being found in the Chicago area around the last week of April and the first week of May. But the exact time varies every year.
Even if you're only interested in buying morels, not in foraging for them, it can be helpful to pay attention to this map's updates. The map lets you know when sellers should begin having them available. Sellers often sell out of them very quickly, so it's important to get there early.
Morel mushrooms are usually only foraged for a 2 to 3 week period every year.