Nightly wagon rides at Pioneer Hall at 7 and 9:30 p.m.- Cost $8 adults, $4 children. Depending on Magic Kingdom hours, there are sometimes special "Fireworks" wagon rides scheduled
or
Private carriage rides - Cost $30/half-hour. Call (407) 824-2832 to reserve (reservations suggested, not required). Depart from Crockett's Tavern at Pioneer Hall between 6 p.m. and 10 p.m. nightly. There are two carriages available; one accommodates four adults and two children, the other accommodates two adults and two children
I do not know about the hay ride, but the cariage ride, you do need to make reservations for.
HTH
Hay rides are nightly. Just go down to the settlement and look for the wagon and the two horses. Opposite of the food. It's first come first serve, and the prices above are correct.
I just read today on the AllEars.net site that you get the carriage rides at Wilderness Lodge, going into the woods area, for $30/carriage and they hold 2 adults 3 children , or 4 adults.