It sounds like it was recorded more than once in the 1920s, in Virginia and North Carolina. The hundred-to-one ratio in the Lomax version is not particularly accurate: according to the site that Bev and Jerry referenced, the machine can produce three to fifteen times as many shoes as an experienced cobbler can in a day. The version I sing goes:

They've invented a machine (peg and awl, peg and awl, etc.)
They've invented a machine:
I peg one, it pegs fifteen --
Throw away my pegs, my pegs, my pegs, my awl.

I'm still wondering when the machine got from Massachusetts to Appalachia and when people started making up songs about it.

By the way, "Long Peggin' Awl" is a completely different song and doesn't belong in this thread.