"sullen," 1744, probably from Old English asolcen "idle, lazy, slow," from past participle of aseolcan "become sluggish, be weak or idle" (related to besylcan "be languid"), from Proto-Germanic *seklanan (cf. Middle High German selken "to drop, fall").
"light carriage with two wheels," 1756, apparently a noun use of sulky (adj.), on notion of "standoffishness," because the carriage has room for only one person.