The following horse carts become available when you complete the indcated task.

Unlock Cardboard Cart:
Raise Oracle's friendship to two Flowers and have your Request Level at rank 4 or higher. Also required is 1 Mythic Ore, 5 Brown Alpaca Wool, and 30,000,000 G.

Unlock Chicken Cart:
Purchased at either Animal shop starting in Year 3.

Unlock Fancy Cart:
Win any Cooking Festival in Year 2.

Unlock Lion Cart:
Purchased at either Animal shop starting in Year 3.

Unlock Shrine Cart:
Win any Cooking Festival in Year 2.

Unlock Sled:
Have the Fancy Cart and Shrine Cart, and have the tunnel between both towns reconnected, then win a Cooking Festival starting in Year 2.

Unlock UFO Cart:
Raise Oracle's friendship to 2 Flowers and have your Request Level rank 4 or higher. You will also need one Stone Tablet, five White Alpaca Wool, and 19,771,116 G.