Good searching.

The reason the song cropped up in 1950 was because Bob Busby, president of the Midwest chapter of the Football Writers Association, sang it to start their meeting. One paper I found mentioned the meeting was held in Kansas City on August 13th, but that seems impossible as the paper was printed on July 26th, 1950 -- unless of course they mean it was the 1949 meeting. I'd assume they meant July 13th? Either way, it was said to "not original with Bob, either" (The Odessa (Tex.) American, July 26th, 1950, p. 7)

The closest I've come to finding the author is a funny epithet that the poem was often printed with: some variation of "'Tis reported that the above message was found on a dying Rutgers halfback." (The Nashville Banner, September 1st, 1950, p. 36)

Also worth noting for searching is that several sources replace "the wily quarterback" with "the cerebral quarterback," however I did find examples of both.