He didn't give any wagons or oxen to the Kohathites because their responsibility was to carry on their shoulders the holy objects assigned to their care.
They placed the Ark of God on a new cart and brought it from Abinadab's house, which was on a hill. Uzzah and Ahio, Abinadab's sons, were directing the cart
When all the elders of Israel had gathered, the priests picked up the Ark and brought the Ark of the Lord and the Tent of Meeting with all its holy items.
Because you weren't there the first time to carry the Ark the Lord our God burst out in violence against us. We didn't treat it in accordance with his instructions.”
Their assigned responsibility was to care for the Ark, the table, the lampstand, the altars, the articles of the sanctuary used with them, the veil, and everything connected with these items.