What does Grandma catch with the cheese in A Long Way From Chicago?
Grandma catches catfish with the cheese.
Grandma catches the catfish in traps, which are actually illegal in Illinois. Nonetheless, she has a series of traps set at various places along Salt Creek, which she baits with gobs of cheese "that smelled bad enough to gas a cat". Mary Alice first discovers the cheese when she goes to the outhouse one morning, and comes out to report that "something died in the cobhouse". Grandma responds curtly that "It's cheese", and when a horrified Mary Alice exclaims that she doesn't want any, Grandma assures her, "It's not for you".
Grandma tells Joey and Mary Alice little more than that they are going fishing, and the three set off with a picnic lunch and the reeking sack of cheese, which Grandma carries herself, "tied to the end of a tree limb hitched up on her shoulder". After they pass a large sign prohibiting fishing, Grandma recovers her sunken traps, which are filled with catfish, "the ugliest thing with gills". After emptying the traps, she baits them with fresh cheese, and lowers them into the creek again.
Grandma's very productive method of fishing is clearly illegal, but she overlooks that fact, most likely justifying her actions because they are for a good cause. Grandma cooks up a feast with the catfish for the down-and-out drifters passing through the area, charitably - and again illegally - running her own "soup kitchen without a license from the Board of Health" ("One-Woman Crime Wave").