Puke
late 16th century: probably imitative; first recorded as a verb in: ‘At first the infant, mewling, and puking in the nurse's arms’, in Shakespeare's As you like it (II. vii. 144).
wiktionary
Probably imitative; or, alternatively from from Proto-Germanic *pukaną(“to spit, puff”), from Proto-Indo-European *bew-(“to blow, swell”). If so, then cognate with German pfauchen, fauchen(“to hiss, spit”). Compare also Dutch spugen(“to spit, spit up”), German spucken(“to spit, puke, throw up”), Old English spīwan(“to vomit, spit”). More at spew.
Attested as early as 1581, first mention is the derivative pukishness(“the tendency to be sick frequently”). In 1600, "to spit up, regurgitate", recorded in the Seven Ages of Man speech in Shakespeare's As You Like It.
This etymology is incomplete. You can help Wiktionary by elaborating on the origins of this term.
etymonline
puke (v.)
"to vomit, eject the contents of the stomach," 1600, probably of imitative origin (compare German spucken "to spit," Latin spuere; also see spew (v.)). First attested in the "Seven Ages of Man" speech in Shakespeare's "As You Like It." Related: Puked; puking.
puke (n.)
1737, "a medicine which excites vomiting;" by 1938 as "material thrown up in vomiting," from puke (v.). U.S. colloquial meaning "native of Missouri" (1835) might be a different word, of unknown origin.
It is well known, that the inhabitants of the several western States are called by certain nicknames. Those of Michigan are called wolverines; of Indiana, hooshers; of Illinois, suckers; of Ohio, buckeyes; of Kentucky, corn-crackers; of Missouri pukes, &c. To call a person by his right nickname, is always taken in good part, and gives no offence; but nothing is more offensive than to mis-nickname—that is, were you to call a hoosher a wolverine, his blood would be up in a moment, and he would immediately show fight. [A.A. Parker, "Trip to the West and Texas," Concord, N.H., 1835]
Bartlett (1859) has "A nickname for a native of Missouri" as the second sense of puke (n.), the first being "A mean, contemptible fellow." The association of the state nickname with the "vomit" word is from at least 1858, and folk etymology talks of the old state literally vomiting forth immigrants to California.
