Yes you can live on a boat in Chicago during the boating season.
Many people all over and around the Great Lakes, and the Hudson, Ohio, Illinois, and Missouri and Mississippi rivers live on their boat during the boating season. Canada in fact has a huge number of live aboard boaters that live full time on their boats during the boating season.
In addition, you have about a hundred or so Great Loopers per season, living on their boats as they make way around the Great Loop.
The "legal" aspect to living aboard is whether or not the body of water is navigable to the ocean. If so, you are good to go. It is against the law however to live on land locked lakes, and waters that do not drain to the sea.
For several years I lived full time on my boat in Buffalo, NY during the boating season, as many others do as well.
John
Oh. . . the only advice I can give you - is get prepared to love it. Once you experience this lifestyle, it is very hard to give it up. That is exactly why you will find me on my boat, in the south during winter, living on my boat full time all year long.