Frank Churchill (1901-1942) composed most of the music for Snow White and the Seven Dwarfs, including "Heigh-Ho" and "Someday My Prince Will Come." The film was nominated for Best Music and Best Score in 1938. Churchill joined Disney in 1930.
(From legends.disney.go.com)

Larry Morey (1905-1971) wrote the lyrics to "Someday My Prince Will Come" and apparently was a sequence director for the film. According to imdb.com, Morey co-wrote Bambi.

Someday My Prince Will Come - Bill Frisell & Fred Hersch
Someday My Prince Will Come - Dave Brubeck Quartet
Someday My Prince Will Come - Iwao san
Someday My Prince Will Come - Eddie Henderson
Someday My Prince Will Come - Snow White
Someday My Prince Will Come - Herbie Hancock and Chick Corea
Someday My Prince Will Come - Bill Evans Trio
Someday My Prince Will Come - Clare Fischer
Someday My Prince Will Come - Nhop Trio
Someday My Prince Will Come - Miles Davis
Teachers Chase - Richard Hewson Orchestra
The Distant Dreamer - Ramsey Lewis
Charade (Main Title) - Henry Mancini
Your Love Won't Turn Me Loose - G.C. Cameron
Moon River - Vince Guaraldi Trio
Secret Life of Plants - Belinda Underwood
I Just Don't Want to Be Lonely - Blue Magic
Festa - Gilfema + 2
Spicks and Specks - Richard Hewson Orchestra
Everybody's Talkin' - Ramsey Lewis
If You Don't Love Me - G.C. Cameron
Wichita Breakdown - Gary Burton Quintet
So Sei Viver No Samba - Cibelle
Riverboat - G.C. Cameron
Upa Neguinho - Jackson Conti