The work is “Little Red Wagon.” written by Richard M. Jones in ~1936. This song is sometimes also known as “Red Wagon.” It morphed into “Your Red Wagon” (a/k/a “Red Wagon” and “Your Little Red Wagon”) credited to Jones, Gene de Paul and Don Ray in about 1947.
The tune itself is pretty “generic,” similar to “Jim Jackson’s Kansas City Blues” (1927), “Mind Your Own Business” (Hank Williams), and “Rock Around the Clock.”
The first commercial release of the Jones version that I have found was by Georgia White, recorded in 1936 and released in 1937 on Decca 7287. A release by Jimmie Lunceford and His Orch. vocal refrain by the Dandridge Sisters dates from 1940 (Columbia 35782). Another cover of the Jones song was by Buddy Jones in 1941.
An early performance of the de Paul/Raye variation was by Marie Bryant in the movie They Live By Night (1948 release) . The Andrews Sisters (Vic Schoen Orch.) (Decca 24268), and Tony Pastor and His Orchestra (Columbia 37973) made commercial recordings in 1947. Later covers include Count Basie (1959) (vocal by Jimmy Rushing), Ella Fitzgerald (1958), Asleep at the Wheel (1985), and Wendi Williams and the Bill Elliott Swing Orchestra from the soundtrack to a TV movie called Introducing Dorothy Dandridge (1999 release). Other covers include Bob Wills (1946), Ray McKinley (1956), and Red Garland (1959, Prestige PRCP 7157.
“Little Red Wagon” (Jones), by Georgia White (1937):
Auto-generated clips here:
http://www.78discography.com/Dec7000.htm
http://rateyourmusic.com/release/single/georgia-white/grandpa-and-grandma-little…
“Your Red Wagon” (Jones), by Jimmie Lunceford and His Orch., vocal by the Dandridge Sisters (1940):
http://www.78discography.com/COL35500.htm
http://www.vocalgroupharmony.com/RedWagon.htm
“Red Wagon” (Jones), by Buddy Jones (1941):
http://www.45worlds.com/78rpm/record/5950
http://countrydiscography.blogspot.com/2010/12/buddy-jones.html
“Your Red Wagon” (de Paul/ Raye), by Marie Bryant from They Live by Night (1948):
“Your Red Wagon” (de Paul / Raye), by the Andrews Sisters (Vic Schoen Orch.) (1940):
http://www.78discography.com/Dec24000.htm
“Your Red Wagon” (de Paul / Raye), by Ella Fitzgerald (1958):
https://www.discogs.com/Ella-Fitzgerald-Travlin-Light/release/8328693
“Your Red Wagon” (de Paul/ Raye), by Count Basie and His Orchestra (1959), vocal by Jimmy Rushing:
https://www.discogs.com/Count-Basie-His-Orchestra-St-Louis-Boogie-EP/release/301…
“Your Red Wagon” (de Paul / Raye) by Tony Pastor and His Orchestra (1947):
https://www.discogs.com/Tony-Pastor-And-His-Orchestra-Gonna-Get-A-Girl-Your-Red-…
[No clip found, although there are some live clips of Pastor performaing this song.]
“That’s Your Red Wagon” (de Paul / Raye), by Asleep at the Wheel (1985):
Auto-generated clips here:
https://www.discogs.com/Asleep-At-The-Wheel-Asleep-At-The-Wheel/release/4653085
“Your Red Wagon” (de Paul / Raye), by Wendi Williams and the Bill Elliott Swing Orchestra (from Introducing Dorothy Dandridge soundtrack):
https://en.wikipedia.org/wiki/Introducing_Dorothy_Dandridge
https://www.amazon.com/Introducing-Dorothy-Dandridge-Original-Soundtrack/dp/B001…
http://www.allmusic.com/album/introducing-dorothy-dandridge-mw0000243663
