We are picking up our car from the Swan this trip (I think it's the Swan, it's either that or the Dolphin) but you can also pick up from the Car Care Centre in MK parking lot (sort of!)
I think there is also a car hire location just up from Downtown Disney - possibly at the Holiday Inn that used to the Courtyard by Marriott? I may be wrong though!
I think both locations are through Alamo - that is who we booked with anyway! For rates etc check pout their website, or do it through Bon Voyage who we booked with. We have a 4 door Intermediate for 7 days fully inclusive of insurances, extra drivers etc for £160.