Wikipedia has interesting articles on Sittin' on Top of the World, and on the Mississippi Sheiks.

Here's what they say about the members of the group:

The Mississippi Sheiks consisted mainly of the Chatmon family, who came from Bolton, Mississippi and were well known throughout the Mississippi Delta. The father of the family, Henderson Chatmon, had been a "musicianer" (someone with good technical ability on his or her instrument adept at sight-reading written music) during slavery times, and his children carried on the musical spirit. Their most famous (although by no means permanent) member was Armenter Chatmon - better known as Bo Carter - who managed a successful solo career as well as playing with the Sheiks, which may have contributed to their success.
When the band first recorded in 1930, the line-up consisted of Carter with Lonnie and Sam Chatmon, and Walter Vinson. Charlie McCoy (not to be confused with Charlie McCoy, a later American musician) played later, when Bo Carter and Sam Chatmon ceased playing full-time. It was Lonnie Chatmon and Vinson who formed the real center of the group.

The Wikipedia article on Sittin' on Top of the World has a photo of the record label, Okeh Cat. No. 8784. Title of the song is "Sitting on Top of the World" by Jacobs-Carter, recorded by the Mississippi Sheiks. The Wikipedia article says the song was recorded February 17, 1930, and the writers were Walter Vinson and Lonnie Chatmon.

Note that the Digital Tradition says the song was written by Lonnie Carter and Walter Jacobs.

The Harry Fox Agency's Songfile.com says the song was written by L. Carter and Walter Jacobs Vinson, Publisher MPL Music Publishing, OBO Edwin H. Morris & Co.

So, the authorship is confusing, but it's clear that it was written by members of the Mississippi Sheiks.

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As for the other "Sitting on Top of the World," there is a fakebook index that attributes the song to McHugh/Koehler, but I think Jim Dixon's attribution (Words, Sam Lewis & Joe Young. Music, Ray Henderson) is more credible.