Most teams stay under issued names such as Union Station Hyatt.
I believe the Ayres Hotel...that is where the Mets usually stay.
There is no telling if the Baltimore Orioles stay in a certain hotel when they are playing in Chicago. There are many hotels in Chicago, so they could stay anywhere from the Holiday Inn to the Drake Hotel.
marriot hotel
Drake
The Peninsula
An average night's stay at the Peninsula Hotel in Chicago costs about $800. There are occasional deals starting at $450 for one night stay. It is a beautiful 5 star hotel.
Westin
The Quality Inn and Suites Anaheim Resort is a great place. It is close to disney land and a fun and clean place to stay.
They stay at their homes in Chicago It is possible that some of them live in hotels
In some hotel, just like every other hockey team on the road.
Assuming you will be entering as a traveler at the city's most popular port, O'Hare, I would direct you to the Chicago O'Hare Extended Stay Hotel. It is custom-made for travelers, and should you be looking for a different location there are dozens elsewhere in the city.