The "blanket organization" for Barbershop in the US (and perhaps some places elsewhere) is the "SPEBSQSA" (Society for the Preservation of Barbershop Quartet Singing in America). They do have a web site, and links there offer at least a few downloadable sheet music samples.

Perhaps if normal search methods don't come up with something, a visit to their site may suggest additional places to look. Most BS groups (quartet and otherwise) include at least a few novelty songs, and this sounds like one that would be popular(?).

John