J.E. Lighter, Random House Historical Dictionary of American Slang (1994) has a couple of citations for "chicken out" that predate the OED's first instance from 1943 (cited in Hugo's answer):
chicken out v. to back out, as from fear; renege. Also (in more recent use) chicken. [First two cited instances:] 1934 Weseen Dict. Slang 177: College {Slang}...chicken out—To fail {sic}. 1941–42 Gach In Army Now 142: Old Uncle Sam promised to let me go back ...in a year. He can't chicken out on me now.
Throughout its period of use in U.S. slang, "chicken out" has coexisted with the adjective chicken in the sense of "pusillanimous." Here is the start of Lighter's entry for chicken in that sense:
chicken adj. 1. cowardly; afraid. [First three cited instances:] 1933 D. Boehm & E. Gelsey Jimmy Dolan (film): Ain't turnin' chicken, are ya? 1929–34 Farrell Judgment Day 532: He was the skinny, dark-haired punk around the corner who was so chicken, wasn't he? 1939 They Made Me a Criminal (film) Boy, is that guy chicken.
A mere 125 years ago, chicken seems not to have carried any sense of cowardice or of failure to follow through. J.S. Farmer & W.E. Henley, Slang & Its Analogues (1891) lists meanings for chicken that range from "a pint pot" (in thieves' cant) to "a child or young person" to an anticipated benefit or success (in the phrase "don't count your chickens before their hatched") to "a poulterer; also a sportsman's term for anyone shooting immature game"(in the term "chicken-butcher") to "any fare out the common, and also to show of any kind" (in the term "chicken-fixings"). And that's it.
The common (until it was banned, in 1835) spectacle of cockfighting in England may help explain why, in Dombey and Son (1846–1848), Dickens identifies a renowned pugilist and athletic adviser to the naive but good-hearted Mr. Toots as "the Game Chicken":
In this delicious abode [his "choice set of apartments"], Mr. Toots devoted himself to the cultivation of those gentle [sporting] arts which refine and humanise existence, his chief instructor in which was an interesting character called the Game Chicken, who was always to be heard of at the bar of the Black Badger, wore a shaggy white great-coat in the warmest weather, and knocked Mr. Toots about the head three times a week, for the small consideration of ten and six per visit.
It turns out that "the Game Chicken" was the nom de guerre of the British boxing champion Henry Pearce in the early 1800s, as described in History of British Boxing, From Fig and Broughton to the Present Time, chapter 2, in The Sportsman's Magazine (July 19, 1845). It should be clear from this example that chicken bore no hint of cowardice or reneging in the 1800s; those associations seem to have developed in the first half of the twentieth century.