Riders consider using horses to pull carts a grave insult to them and the goddess Gamari. Therefore, they use travoises, triangular structures of two long poles and a surface or set of shorter poles strung between them, instead.
Although the Wheels consider their chariots far superior to a travois, it's worth noting that travoises do better on uneven terrain than a wheeled equivalent!
(Rams have neither horses nor theological debates about them; they use carts pulled by oxen instead. The exception to this is the Berenethtelli in Lights Going Out, who carry goods by travois in the Rider tradition, same as your clan in Ride Like the Wind.)