The Illini live on today in the Confederated Peoria Tribe of Oklahoma, who have a shared language and culture with the Miami (or Myaamia) people. The Myaamia Center at the Miami University of Ohio (myaamicenter.org) has sponsored some research in exactly this area.

They cultivated and consumed a lot of corn (maize), hunted and preserved buffalo and other game, and gathered a wide variety of local roots, fruits, and vegetables, depending on what was in season. The people that lived there had a great depth of knowledge of the local flora, which parts were edible, and how to prepare it. Some plants are poisonous if not prepared correctly, and some "poisonous" plants may have certain parts be briefly edible during certain times of year.

These tribes also had relatively wide-ranging trade contacts, particularly along major rivers. I don't know how much actual "food" moved along these routes, but it would not be inconceivable for highly desirable items to travel this way. As long as it doesn't spoil too quickly. And mostly for rich people (aka "high status individuals").

A random 21st century civilized person would probably require a local to help them find and prepare much food. Also, absent widespread social disruption (where whole tribes flee, and the land becomes uninhabited), wherever they went they would be on somebody's "turf". Who might be disappointed to find a stranger camping in the middle of their favorite pecan grove, much less raiding their cornfields.

Sources:

Gonella, Michael Paul. Myaamia Ethnobotany. Diss. Miami University, 2007.

King, Brian. The Ethnobotany of the Miami Tribe: Traditional plant use from historical texts. Diss. Miami University, 2003.

Ironstrack, George Michael. From the Ashes: One Story of the Village of Pinkwi Mihtohseeniaki. Diss. Miami University, 2006.