There are a couple of threads (
here, for example) about Chicago food gifts and local foods elsewhere, but nothing with quite this focus. If you were putting together a care package of groceries for somebody who's moving away from the Chicago area, what would you put in it that might be tough to obtain elsewhere?
Refrigerated products are OK, but not highly perishable ones. (That is, things that have to be packed in a cooler are fine; those needing dry ice or which won't keep for more than a week or so, probably not. So meat products are all right -- they can be frozen on arrival. Eli's Cheesecake maybe. Village Creamery ice cream, sadly, won't work.)
They don't need to be foods made in Chicago, just stuff a Chicagoan moving to a small- to middle-sized Midwestern city would have trouble getting and would miss. Here are a few of the items I've thought of so far:
Giardiniera
Neon green relish
Vienna natural-casing hot dogs
Maryann poppyseed buns
Celery salt (OK, you can get that anywhere, probably but it goes with other stuff.)
Intelligentsia coffee
Sausages and cured meats from Bobak's, Bende and Roumanian Kosher
Vitner's potato chips
Kimchi
Gojuchang
Local barbecue saucesBagels from New York or Chicago Bagel & Bialy
Belly lox
(They aren't beer drinkers, so no local beers. Maybe a bottle of Malort, just for fun.)