Thanks to a helpful Reference Librarian in a library far, far away:
(Transcribed from a PDF of the above excerpt)
The passage preceding the lyrics refers to Gene Autry and Bradford Browne:
"Gene likes and sings Brad's "Git Along Little Pony" best but we like his "Cowboy Without a Guitar" best."

Cowboy Without a Guitar   by Bradford Browne

One day in Wyomin' I met an old man,
For years he'd been searchin' for gold
He told me a story as strange as could be-
And this was the story he told.

I have traveled this mighty world over,
From Cheyenne to old Zanzibar.
'Twas in Texas that I saw the strangest of sights,
A COWBOY WITHOUT A GUITAR.
He was riding a bucking young bronco,
Who tossed him about near and far.
I had to look twice to be sure that I saw
A COWBOY WITHOUT A GUITAR.

CHORUS:

Oh, sing me a song of the West, Mister Cowboy,
Yippi I, yippy ay, git along little stray.
Oh sing me a song of the West, Mister Cowboy,
And then I'll be off on my way.
I've seen coyotes that howl in the daytime
And a monkey that smoked a a cigar,
But I ain't been the same since the day that I saw
A COWBOY WITHOUT A GUITAR.

I thrill at the sight of Niag'ra
My heart jumps at each movie star,
But the one thrill I got that I'll never forget
A COWBOY WITHOUT A GUITAR.
He was brandin' a steer in Montana,
In his hand was a smokin' red bar,
At first I was doubtful, but folks, there he was,
A COWBOY WITHOUT A GUITAR.

CHORUS:

Oh, sing me a song of the West, Mister Cowboy,
Yippi I, yippy ay, git along little stray.
Oh sing me a song of the West, Mister Cowboy,
And then I'll be off on my way.
I've seen princes and kings in pajamas,
And rustlers in feather and tar,
But I'm ready to die now that I have beheld
A COWBOY WITHOUT A GUITAR.

Found in "How daddy became a beachcomber", by Marilyn Hedley, pg' 96-97.

Perhaps Ebbie's friend or someone else can reconstruct/locate the tune, if the sheet music doesn't emerge soon. I did alert Ebbie to this revived thread.