The Hoagie Hut at the corner of Lincoln and Wrightwood serves what's called a "Detroit Dog" on the menu. Ground beef in sauce, mustard, onions on a dog.
However, Coney buyer beware: when my fiancee, a lifelong Michigander, ordered the Detroit Dog, the counter help shuffled around for a few minutes and finally had to ask her how exactly one was made. Guess they hadn't had anyone order one of those for quite a while.
That might be the closest you can get in the city. I know that Happy House in West Chicago serves a Sloppy Joe Dog, which one could concievably order with mustard and onions as well. A sad facsimile, to be sure, but necessity is sometimes the mother of improvised dogs.
Kicked while you're down: Just today had a couple coneys from Tom's Coney Cafe outside Flint, MI. One full coney, and my personal favorite--loose meat, no dog, in the bun, no onions. Mmmm. Also, there's 48 cans of Faygo Rock N' Rye in my trunk, and a package or two of Kogel's Viennas in the cooler. If only I had thought to pick up some pasties on the way back from the Platte River...
Hoagie Hut
2580 N Lincoln Ave
Chicago, IL 60614
Happy House
140 East Roosevelt Road
West Chicago, Illinois 60185
Tom's Coney Cafe
6339 S Saginaw Rd
Grand Blanc, MI 48439