I was in the same boat as you for a few months: lived in ORD but was based in DEN. Unfortunately, the company only provides you the $36 stipend for parking, and you're on your own to find somewhere to park. I used a combination of the off-airport private parking lots (Pre-Flight Parking was my favorite) and getting rides from friends and family, as I don't live near any public transportation.

The parking lot is the AMC lot at Montrose and Mannheim on the east side of the airport. There is a shuttle that runs every 10-15 minutes and drops off and picks up at the departure level of every terminal 24 hours a day. Once you're based in ORD, the CPO can issue you an AMC parking pass, but it seems that these passes are issued directly to the company and there is no way to get one just for yourself - even if you pay for it on your own (the lot isn't associated with the ORD badging office and I couldn't find an "office" or point of contact for it).

Once you get awarded ORD, go up to the CPO and ask if you can have the badge then. It worked for me and I got about a month of parking in the AMC lot before I was officially based there.

I hope that helps, and feel free to PM with any questions.