I know the Texas Rangers and the Mil. Brewers stay at the Westin on Michigan Ave. I would suspect that is where all of the out of town teams stay.
Where all the major league teams stay when they play the RAYS- The Renaissance Vinoy Hotel in Saint Petersburg!
The Chicago Blackhawks stay at the Island Hotel in Newport Beach.
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
Either the Ritz Carlton or the Arizona Biltmore
They stay at their homes in Chicago It is possible that some of them live in hotels
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.
Park Hyatt Hotel