Short answer - Nope. Within the city, there is Chicago Xplosion. They are way South - maybe 75th St I think? I haven't seen them in awhile, so I don't know if they still exist. And a quick Google search has turned up something called Cheer Destiny in Lincoln Park. I've never heard of it before now, so maybe it's new? It says they have allstar cheerleading on their website. Maybe look there.
I live about 10 minutes from where you're talking about in Lincoln square.
There are about a bazillion gyms in the various suburbs of Chicago. Getting to the suburbs of Chicago on weekday afternoons requires the patience of Job, but can be done.
Just know that any gym you find in the suburbs, add 45 minutes to whatever Google maps says, because most of the time that's what will happen. It says my cp's gym is just over an hour away, I don't think we ever got there in less than 2 hours.
If you end up in the gold Coast area, despite what a map tells you, South suburbs will probably be your best bet, traffic wise.
Don't know if this helps - if you do end up moving, I can be more help once you know where. There are lots of kids activities in the city - but cheerleading is NOT popular in Chicago. I still get "the look" (confusion, eye rolling, the "Oh........ OK....... I guess.....") and my kids haven't cheered in 2 years.
(Btw - if you're definitely moving to the city and not the suburbs, I'd be more worried about schools than cheer. Learned that lesson the hard way when we moved here. The schools here require patience and diligence - it's an elaborate system. If you move here and don't go with private school, I can help you navigate the school system too - I'm a pro)