Found it! The sheet music can be seen here: at Indiana State University.


HORSEY, KEEP YOUR TAIL UP (KEEP THE SUN OUT OF MY EYES)
By Walter Hirsch and Bert Kaplan
M. Witmark & Sons, 1923

VERSE 1: Reuben is so silly, he will never wear a hat.
Tho' the sun is shining, he don't give a darn for that.
You can see him driving in his buggy ev'ry day.
He just leans back and shakes the reins, and you can hear him say:

CHORUS 1: Oh, Horsey! keep yer tail up, keep yer tail up, keep yer tail up.
Oh, Horsey! keep yer tail up. Why don't you make it rise?
Tell me how you get that way.
You mean "Yes" and say "Neigh, neigh."
Oh, Horsey! Keep your tail up! Keep the sun out of my eyes!

VERSE 2: Ev'ry Sunday morning Reuben goes out for a ride,
Ev'ry Sunday morning with his girlie by his side.
If the day is cloudy, ev'rything is very well,
But when it's bright, it's his delight to get right up and yell:

CHORUS 2: Oh, Horsey! keep yer tail up, keep yer tail up, keep yer tail up.
Oh, Horsey! keep yer tail up. Why don't you make it rise?
Keep it up or off it goes.
Horses' tails make fiddle-bows,
So, Horsey! Keep your tail up! Keep the sun out of my eyes!

CHORUS 3: Oh, Horsey! keep yer tail up, keep yer tail up, keep yer tail up,
Oh, Horsey! keep yer tail up. Why don't you make it rise?
Keep it up, I'll lose my breath.
Gosh! you tickle me to death,
So, Horsey! Keep your tail up! Keep the sun out of my eyes!