Bowling Green State University has the sheet music. It's not viewable online, but their catalog has this information:
MISTER SEARS AND ROEBUCK
Words by Ray Gilbert and William Okie ; music by Al Gannaway.
Hollywood, CA : Mark Warnow Music, ©1949.
"Recorded by Dorothy Shay on Columbia Records"
I consider the above information to be definitive, although some other sources say the title is "Mr. Sears & Mr. Roebuck" or other variations.
By the way, I listened to Dorothy Shay's recording on Spotify, and her lyrics correspond EXACTLY to those I posted back in 2003, which I copied from another website.