A sulky person is one who, at least for the moment, wants nothing but to be let alone, and it is directly from this meaning that the vehicle sulky gets its name.
For this vehicle is designed to seat a single person, and one choosing to ride therein presumably feels that degree of aloofness best described by the adjective giving it its name.
The same line of reasoning has led to the application of the same name to such things as a bathing machine designed for one, a horse-drawn plow having a seat for a single rider, etc.
