I'm the warehouse manager. When submitting the purchase requisition to the purchasing department, how to effectively and better urge the purchase to arrive as soon as possible?

As a warehouse manager, it seems that you are not only responsible for keeping the materials, but also for expediting the materials you mentioned. In order to complete this work better, because the procurement management authority and process of a unit and a unit are different, you must master two time periods, one is the specific use time of the user department, and the other is the procurement cycle (including transportation) of each material. If the procurement cycle is not enough, neither you nor the procurement department is responsible. If the procurement cycle is sufficient, the procurement department is responsible. It is best to let the user department put pressure on the purchasing department on some occasions, and your life will be easier.