The sawmill can stop for a few reasons.
There are two warehouses in the game: the Sawmill Warehouse for wood materials and the Workshop Warehouse for building materials. Make sure you are checking the correct one.

Production stops if the Sawmill Warehouse is full. Complete some orders, turn on Samuel’s auto orders, or use a Flash Sale to free space.

A stuck order can also block progress. This happens when the order requires materials you do not have. Cancel the order and choose one that matches what is in your warehouse.

Finally, check that the production line matches the type of product required by the active order. If they do not match, the order will not move forward. For example, if the order needs cants but your warehouse contains only rough lumber, the order cannot be completed. Cancel the order and choose one that matches the materials currently in your warehouse.