Can you transport a new in the box washing machine laying on its back?

It is not recommended. The drum and motor assembly of the typical top-loading automatic washing machine are suspended from the sheet metal cabinet in a way that allows them to move when the load is out of balance. The suspension mechanism is designed to hold the weight of the drum, agitator, motor, pump, water and clothes in a vertical position, and if tipped on its side the mechanical parts may be damaged in transit, unless they are first blocked with padding material to keep them in the center of the cabinet. If the new-in-box machine has instructions on it to the effect of "THIS SIDE UP", heed the instructions or risk damaging the machine and voiding the warranty.