"Shine on, Harvest Moon" was written in 1908 by Nora Bayes and Jack Norworth, a married composing team that predated Goffin and King and Weil and Mann by half a century. It premiered in the Ziegfeld Follies and quickly became a vaudeville favorite. It has endured as a standard for over 100 years, perhaps having originated the euphemism "spoon" for not quite you-know-what (a year before "Silvery Moon;" see bottom of the post).
A "harvest moon" is simply a full moon in September or October, presumably allowing work in the fields at night, a reference that would have been immediately recognized in more agricultural times.
Very early 1908 recording by Harry MacDonough and Elise Stevenson (as "Miss Walton"):
Bonus (h/t W.E.):