Please process the shipment if the inspection is passed
If there is a process to follow, this is possible for the receiver. (They receive the shipment and then 'process' it.) Some people might use the expression 'process the shipment' meaning 'prepare the necessary constituents, ready for shipment', but in my view a 'shipment' doesn't exist until it's been shipped.
Please proceed WITH the shipment if the inspection is passed
The second version would apply to the shipper, and needs a 'with'.