From Brewer’s Dictionary of Phrase and Fable: “E”

FROM BREWER’S DICTIONARY:

Eyes out on the stalks – Eyes figuratively or even actually protruding through inquisitiveness, amazement, fear, lust or other strong emotion or reaction. The allusion is to the eyes of a snail, which are at the end of retractable stalks and suggest alertness, The phrase dates from the 1930s.