Some time back Disney was offering special "up-close" theme park parking to guests who rented from one specific car rental company, as part of a marketing tie-in. Do they still do this and if so...what car rental company?
Haven't heard about any special parking except for the AAA Diamond parking at each park. But if I had to guess, I'd say look into Alamo, as they are the official car rental company of WDW, as well as the sponsor of the TTA.
Best hold on to Your Hats and Glasses, 'cause this heres the Wildest Ride in the Wilderness!
Way back when, in ye olden days, Disney had an association with National and actually had a section at their lots for National car renters, something we took advantage of one time we were there. Truth be told, it wasn't really all that much closer to the park and after the promotion ended we found that getting there early with our own car was just as good.
As far as I know the only option left is the AAA Diamond lot for AAA card members who book their trip through AAA but you definitely want to check with Disney and/or AAA to see if that is even still available.