According to Warner Chappell Music, there were 2 hit recordings of PLAYMATES in the same year, 1940: One by Kay Kyser and His Orchestra, and one by Mitchell Ayres And His Orchestra. The first recording was apparently by Hal Kemp and the Smoothies; Saxie Dowell, the songwriter, was a sax player (naturally) in that band.
A version performed by the Kipper Kids was used in the soundtrack of "The Addams Family" in 1991 and appears on the soundtrack album (it's only 25 seconds long).
Following are lyrics from Lyrics Playground, although I don't know how much they can be trusted, since they have the title wrong.
OH PLAYMATE, COME OUT & PLAY WITH ME (Words and music by Saxie Dowell) The Playmates
There's a catchy little tune a-floatin' through the air. You hear it here and there. They sing it ev'rywhere. How it started, where it started, seems nobody knows. But what's the diff'rence where it came from? Here's the way it goes:
Oh playmate, come out and play with me And bring your dollies three. Climb up my apple tree. Look down my rain barrel. Slide down my cellar door, And we'll be jolly friends forever more.
It was a rainy day. She couldn't come out to play. With tearful eyes and tender sighs I could hear her say:
I'm sorry, Playmate, I cannot play with you My dollies have the flu, Boo-hoo hoo hoo hoo hoo. Ain't got no rain barrel, Ain't got no cellar door But we'll be jolly friends forever more.