Chicago is a great town for "garage" theatre. You should have no trouble finding something or starting something.
There used to be a paper called "The Reader" I don't know if it's still published, but if it is, "The Reader" publishes a complete listing of everything being produced. You could probably get a feel for what companies are producing puppet shows.
I would recommend placing an ad if you want to start something. Something along the lines of "Seeking collaborators - puppeteers to discuss starting a new troupe." There is so much talent in Chicago, you'll probably be flooded with calls. Meet the ones who sound the most earnest at the local Starbucks and forge your plan for world domination! Er - I mean for a puppet show.
Good luck!
(And if you run into Bill Burman, tell him Andy says "hi.")
-Scraps