Middle English fecchen, from Old English feccan "to bring, bring to; seek, gain, take," apparently a variant of fetian, fatian "bring near, bring back, obtain; induce; marry," which is probably from Proto-Germanic *fetan (source also of Old Frisian fatia "to grasp, seize, contain," Old Norse feta "to find one's way," Middle Dutch vatten, Old High German sih faggon "to mount, climb," German fassen "to grasp, contain"). This would connect it to the PIE verb for "to walk" derived from root *ped- "foot."
With widespread sense development: to "reach," "deliver," "effect," "make (butter), churn" (19c.), "restore to consciousness" (1620s), also various nautical senses from 16c.-17c.; meaning "to bring in as equivalent or price" is from c. 1600. In 17c. writers on language didn't derive a word's etymology; they fetched it. As what a dog does, c. 1600, originally fetch-and-carry. Variant form fet, a derivation of the original Old English version of the word, survived as a competitor until 17c. Related: Fetched; fetching.