How to deal with purchase delays when you have pressure of requests from various projects of the same company?

Examine the use of risk developed by the project man 

Prioritization of requirements by the expected loss due to the delay in the supply of the largest in the loss   to the lowest in the loss. 

Change the shipping method and itineraries in order to achieve the speed of supply, thereby reducing potential losses.

thats depends on your high relation ship with several suppliers so no one can make delay " by mean delay or submission " , never default if you want to be listend every issues 

Prioritize all requests on the basis of their criticality, then take the internal customer on board and decide what to take on first

identify cause of procurement delay.

Qualify between urgent and important. Prioritize important over urgent Requirements.

establish cut off time and date.

Request from engineering or operation department material delivery schedule.