You should arrange this at the time you make your reservation, not when you pick up your car.
The following list of car rental offices on or near WDW property was copied from another post here on the DIS. As you can see, you don't have to limit your shopping to just Alamo/National:
Alamo; At Car Care Center; M-F 6am-8pm, Wknd 6am-6pm; 407-824-3470
Alamo; In Buena Vista Palace; 8am-4pm; 407-827-6363
Alamo; In Dolphin Resort; 8am-4pm; 407-934-4930
Avis; In Hilton at WDW; 8am-6pm; 407-827-2847
Budget; In Doubletree; 8am-5pm (closed holidays); 407-827-6089
Dollar; In Regal Sun Resort; 9am-4pm; 407-583-8000
Enterprise; In Marriott World Center; M-F 9am-5pm, Wknd 9am-1pm; 407-397-0800
Hertz; In Shades of Green; 8am-5pm; 407-938-0600
Hertz; In Caribe Royale; 8am-6pm; 407-239-2502
National; At Car Care Center; M-F 6am-8pm, Wknd 6am-6pm; 407-824-3470
National; In Buena Vista Palace; 8am-4pm; 407-827-6363
National; In Dolphin Resort; 8am-4pm; 407-934-4930
Thrifty; In Radisson Celebration; 24hr; 407-390-1596