Middle English sēchen "go in search or quest of; strive for, try to attain," from Old English secan, seocan "search for; pursue, chase; long for, wish for, desire; look for, expect from," influenced by Old Norse soekja, both from Proto-Germanic *sokjanan (source also of Old Saxon sokian, Old Frisian seka, Middle Dutch soekan, Old High German suohhan, German suchen, Gothic sokjan).
This is reconstructed to be from PIE *sag-yo-, from root *sag- "to track down, seek out" (source also of Latin sagire "to perceive quickly or keenly," sagus "presaging, predicting," Old Irish saigim "seek"). The natural modern form of the Anglo-Saxon word, had it not been influenced by Norse, is in beseech. Related: Sought; seeking.
By late Old English as "ask a question." Seek-sorrow (1580s) was an old term for "a self-tormentor, one who contrives to vex himself." Seek-no-further (or farther) as the name of a type of eating apple is by 1660s.