From the earlier quots. it would seem that box was here originally used in the sense seated compartment, rather than applied to the receptacle under the drivers feet, as stated in the quot. from Chambers 1753 under BOX 6.
The seat occupied by the driver of a coach.
1651. Cleveland, Poems, 28. Resign thy Coach-box Twisse.
1670. Cotton, Espernon, III. XI. 560. They first pulld him [the coachman] out of his Coach-box.
1728. Vanbr. & Cib., Prov. Husb., I. i. Doll puked a little with riding backward, so they hoisted her into the Coach-BoxAnd then her Stomach was easy.
1801. W. Felton, Carriages, I. 125. A handsome Coach-box is a great ornament to a Carriage.
1841. Marryat, Poacher, xv. The guns were unstrapped from the back of the coach-box.
attrib. 1705. Lond. Gaz., No. 4177/4. A blue Furbelow Coach-Box Cloth.