Among vehicles capable of transporting a 4'x8' sheet of plywood, which has the best fuel economy?

I assume you haven't considered a trailer pulled by an economy car (tow ratings vary but they are usually either 1000lbs or zero).

The few minivans that will fit a 4x8 sheet (only a few of the stretched models) are more fuel efficient than the full size pickups that are the other option (no mini, or mid-sized truck has an 8' long bed except some really old Dodge Dakota models)

So it would be a 6cyl full size pickup, which are all very close in mileage.

A Prius Wagon works great. Fold down the rear seat, put the plywood in diagonally, and let it hang about 2 feet out the back. 5 people with the seat up.