Cars were a common theme in early Beach Boys songs, notably "Little Deuce Coupe," which was the template for this song that envisions Santa's sleigh as a Nordic Hot Rod: candy apple red with a 4-speed stick. Brian Wilson wrote the song with Mike Love.
For The Beach Boys' Christmas Album, the first side was original Christmas songs written by Brian Wilson and Mike Love, and the other side contained traditional favorites like "Frosty The Snowman" and "White Christmas." (thanks, Bertrand - Paris, France)
The "Run run, reindeer" line was copped from Chuck Berry's song "Run Rudolph Run," where Berry sings, "Run run, Rudolph." By making the reindeer generic, The Beach Boys avoided copyright issues - Berry had to turn over royalties to his song to the creator the Rudolph the Red-Nosed Reindeer story.
This was used in a popular Coke commercial featuring a moment of Christmas cheer between animated polar bears and penguins.
Hanson recorded this for their 1997 Christmas album Snowed In.