Who knows how far back the chain of inspiration and influence goes, but Robert Lewis May’s book predates Gene Autry [1]. The wiki link you shared actually mentions the inspiration for Gene was a children’s book.
>The wiki link you shared actually mentions the inspiration for Gene was a children's book.
And even mentions May, specifically:
>In 1939, Marks' brother-in-law, Robert L. May, created the character Rudolph as an assignment for Montgomery Ward, and Marks decided to adapt the story of Rudolph into a song.
And even mentions May, specifically:
>In 1939, Marks' brother-in-law, Robert L. May, created the character Rudolph as an assignment for Montgomery Ward, and Marks decided to adapt the story of Rudolph into a song.