The verb of which kempt is the past participle is kemb, “to comb,” and it derives from the Anglo-Saxon cembam and ultimately from the Old Teutonic kembjan.
The only modern survivor of the verb is this form as it appears in the combinations unkempt and well-kempt, but the meanings most often implied in modern usage are, respectively, “ill-groomed, slovenly” and “well-groomed, neat,” rather than the literal meanings of “uncombed” and “neatly combed.”
In all other uses, kemb has been replaced by its descendant, comb.