The soldiers were more independent than in other armies(for food) but they would still have a huge empire that supplied them.
Soldiers would have multiple horses and they could use the milk from them to survive and if necessary even blood.

The army had an organized supply train like any other army.This would carry weapons(mostly arrows) but also food for their animals.

Also, mongol armies were fast.They traveled light and they could move a lot further than other armies in a short period of time.
They would always plan out their attacks, gather information and scout so that also helped with their logistics.