It's possible that "The Bull Frog", in the DigiTrad @displaysong.cfm?SongID=6010 be a source for this children's rhyme.

But I think that this archived Mudcat thread provides what is almost a sure bet as the source for this children's rhyme: Lyr Req: a big bullfrog jumped into the lake

See the following two posts from that thread:

Subject: RE: Lyr Req: a big bullfrog jumped into the lake
From: GUEST
Date: 28 Dec 04 - 06:39 PM

Not really Pete Seeger's, though his version (as "Foolish Frog") sure is catchy. It originated back around the turn of the (19th-20th) century as "May Irwin's Frog Song," one of several of this massively built entertainer's hits— others included "Lulu" and "Bully of the Town." She had a knack of picking up song material from black sources, so it's not impossible that hers are rewrites of even earlier stuff. Irwin repays study; I only wish she'd recorded so we could hear the voice that tickled thousands in vaudeville days. --Bob Coltman

-snip-

Subject: Lyr Add: MAY IRWIN'S FROG SONG
From: Jim Dixon - PM
Date: 01 Jan 05 - 03:49 PM

This is the song mentioned above, but it probably isn't the song littledaddy1803 wanted.

The sheet music for MAY IRWIN'S FROG SONG may be found at The Library of Congress American Memory Collection. May Irwin (1862-1938) was the performer, not the songwriter.

MAY IRWIN'S FROG SONG
Charles E. Trevathan, 1896

1. Away down a-yonder in Yankety Yank,
A bullfrog jumped from bank to bank
'Cause there wasn't nothin' else to do.
He stubbed his toe an' in he fell,
An' de neighbors all say dat he went to well,
'Cause he hadn't nothin' else to do.

CHORUS: An' jus' lots uv folks is like dis foolish frog uv mine,
A-runnin' into trouble jus' to pass de time,
An' de devil's allus loafin' 'round heah jus' to grab de kind
Dat nevah hasn't nothin' else to do.

2. When dey buried dat frog, de preacher said,
"De reason why dis young frog is dead,
'Cause there wasn't nothin' else to do.
An' all you frogs jus' a-listen to me:
Yo' bettah stay at home wid yo' family,
When you haven't nothin' else to do."

3. Some frogs I know is pow'ful fond
Uv spendin' dey time in 'nother frog's pond
'Cause dey hasn't nothin' else to do.
But dis consolation de good book brings:
De frog uv dem habits won't wear no wings
'Cause he hasn't nothin' else to do.

4. Now all uv yo' people dat heah dis song,
Yo' knows shy dis po' frog went wrong:
Cause he hadn't nothin' else to do.
You'd bettah keep busy on any kind of pay,
Till de big horn blow on de judgment day.
Den you will hab somethin' else to do.