I agree that it would not be a small operation. 4780 tons of gold is almost 349 000 lbs of gold. Assuming 1000 lbs/pallet to move, and 16/pallets to a truck, it would take about 22 trucks to move. That’s an average of one night in a warehouse.
Less if they were able to move it larger amounts at a time.
Your figures are a bit off. The average tractor-trailer can haul over 40,000 lbs. of cargo weight and the individual pallets can (and likely do) weigh in excess of 2000 lbs. So 9 truckloads could empty the vault, which could be done in less than an hour.