First, keep in mind that the game uses two different warehouses. The Sawmill Warehouse stores wood materials such as rough lumber, lumber, and cants. The Workshop Warehouse stores house-building materials. Make sure you are looking at the correct warehouse for the part of the game you are playing.

Your Workshop forklift can get stuck for a few reasons:

1) The Workshop order isn’t first in the queue.
The forklift only delivers materials if the order from the man in a helmet is the first active order. If other orders are ahead of it, cancel them.

 
Stuck Workshop order
 
2) The Workshop order requires materials you can no longer produce.
All wood types (cants, rough lumbers, lumbers) are valid, but sometimes the order becomes stuck. Cancel it and accept a new one — the new order will request materials you currently produce.
 
Stuck workshop order
 
3) Your storage is full.
If the total required materials exceed your warehouse capacity, the forklift cannot deliver anything. Either cancel the order or increase capacity by upgrading Ms. Karen.
 
Reached the maximum storage capacityUpgrade Ms. Karen