The car rental and Car Care Center are irrelevant. You're not just staying in Bay Lake. (Bay Lake and Lake Buena Vista are two towns that comprise Walt Disney World.) You're staying at Disney's Bay Lake Tower, the DVC component of Disney's Contemporary Resort. I assume you're staying with DVC points.
Call DVC Member Services. Have your outbound flight information. Book one-way Disney's Magical Express transportation for your check-out date. Do it now, before you start traveling; that way you'll be in the system and you'll have one less thing to worry about on vacation.
You'll get a letter with your pick-up time the day before your check-out. It will be approx. 3 hours before your scheduled flight departure.
If you airline participates in Resort Airline Check-in (all major airlines with flight to Chicago participate), be sure to take use this incredibly convenient "curbside skycap" service to check your bags and get your boarding passes at your resort.