early 15c., "to disturb, trouble," from Middle French soliciter (14c.), from Latin sollicitare "to disturb, rouse, trouble, harass; stimulate, provoke," from sollicitus "agitated," from sollus "whole, entire" (from PIE root *sol- "whole, well-kept") + citus "aroused," past participle of ciere "shake, excite, set in motion" (from PIE root *keie- "to set in motion"). Related: Solicited; soliciting.
Meaning "entreat, petition" is from 1520s. Meaning "to further (business affairs)" evolved mid-15c. from Middle French sense of "manage affairs." The sexual sense (often in reference to prostitutes) is attested from 1710, probably from a merger of the business sense and an earlier sense of "to court or beg the favor of" (a woman), attested from 1590s.