Here's what Pete says in Where Have All the Flowers Gone? a Singalong Memoir (revised edition, 2009)

It was around 1961. Singing in Wisconsin with my younger brother and sister, Mike and Peggy, I found the next verses on the back of the menu of a roadside diner. I had never seen them before except for the first two lines, which I had once seen scrawled on the door of a public toilet.
When I put a tune to it and added a couple lines, I wrote to the hash house. I had stolen their menu. They told me they got the words from a newspaper column in a Milwaukee paper. I wrote to the columnist; he couldn't remember who had sent it to him.
I have since found that the poem has been widely reprinted in different versions. Nobody knows for sure who originally wrote it. If could be from anywhere in the late 19th or early 20th century. I heard from people who remembered it from before World War I. It's sometimes titled "I'm Doing Quite Well For the Shape I'm In."
All I contributed besides the melody were a couple of lines and the idea of repeating the first four lines as a chorus. You'll find that you can do this with many poems that were written originally to be read but need more repetition if they are going to be successfully sung.

We get a lot of posts like the one from D.E. Shiveley above. Are they believable? Here's something else, from www.debrashiveleywelch.net (apparently the same person):

Debra: I am 53 years of age and have been working since age eleven. I grew up very poor and am a veteran of child abuse. I left my home at age 17, three hours after graduation, and entered the work force full-time to take care of myself and to take control of my environment. At age 28 I finally took my first college course and after four years of night school graduated with a degree in Business Management. Okay, so I'm also very practical. My grandfather was a poet and wrote "My Get Up and Go” and many other works, under the pen name of Proxy, that were published in local newspapers in Southern Ohio. My father also wrote poetry as did my mother. I began to write at age nine and have never gotten over the addiction! When I'm on a roll, I can make you laugh until your sides split, I'm a sucker for anyone who is crying and there are times when watching a flock of geese take off from the lake, or seeing the sun play in the fountain sprays, I simply cannot breathe. I am now married 19 years; I love to cook; I cry when I hear bagpipes or hear a singer hit a "glory note” and I think God made old movies, tomatoes and snap dragons just for me!

I know there are things that I believed were written or spoken by one of my ancestors, but it turned out they came from a completely different source. Maybe Debra Shiveley is correct - but that's something for her to settle with Pete Seeger.
-Joe-