RangerSteve, according to google, Captain Kangaroo was one of those who did the song. The site gives only the beginning:

"Then there was Willie the Spider, who had so many legs that by the time he tied
all his shoes, it was time to take them off again and go back to bed. ... "

If one kept digging, you might be able to find the whole song.

(Including this post, I am now at the total of 3, 143!)