Old English huntian "chase game" (transitive and intransitive), perhaps developed from hunta "hunter," and related to hentan "to seize," from Proto-Germanic *huntojan (source also of Gothic hinþan "to seize, capture," Old High German hunda "booty"), which is of uncertain origin.
Not the usual Germanic word for this, which is represented by Dutch jagen, German jagen (see yacht (n.)). General sense of "search diligently" (for anything) is recorded from c. 1200. Related: Hunted; hunting. To hunt (something) up "search for until found" is from 1791. Parlor game hunt the slipper is attested from 1766.
1853 in the literal sense (witch-hunting is from 1630s), from witch (n.) + hunt (n.). The extended sense is attested from 1919, American English, later re-popularized in reaction to Cold War anti-Communism.
Senator [Lee S.] Overman. What do you mean by witch hunt?
Mr. [Raymond] Robins. I mean this, Senator. You are familiar with the old witch-hunt attitude, that when people get frightened at things and see bogies, then they get out witch proclamations, and mob action and all kinds of hysteria takes place. ["Bolshevik Propaganda," U.S. Senate subcommittee hearings, 1919]