http://www.airportexpress.com/ -You must bring your own car seat.
The fee should be about $25 per person (not a great deal, most likely the other options listed below are much less)
At the airport you can catch an American Taxi to the city. They do not require a car seat so you can ride Kazakh style. The fare should be about $40
You can take the L which you catch right at the airport and follow the instructions below:
You would get on the Blue line at O’Hare heading southbound to downtown Chicago.
You would need to transfer from the blue line to the red line at Jackson. This is a free transfer. Once on the red line, take it to Roosevelt. Exist at Roosevelt and then walk 4 blocks north to Michigan & 8th.
Here is a link to the cta system maps
the fee is aprox. $8
Lastly, if you are a high roller you can google O'hare and Midway limo companies...almost all will come with a carseat if requested. The fee for this is aprox. $60-$80 one way.