I think I found a good solution on a 2014 post on the Tesla.com forums.
O'Hare Personal Valet
I called the owner, Glenn, and he said they will charge a vehicle for a flat rate of $30. That comes out to about $30 a day on my five day trip, and $22 per day for my nine day trip. O'Hare valet runs about $53 a day.
I am also looking at one other option on plugshare, preflight parking ORD, which has a row of 16 EV spots. That comes in at $12-$14 per day, but would also include a shuttle transit to terminal vs. just dropping the car off with Glenn's team and walking in.
I will probably try out the personal valet on my first trip, which is the shorter five day trip.