I hear kids singing this all the time in the schools in NY, PA, & OH, and here's what I usually here:

Rudolph the Red-Nosed Reindeer (reindeer)
Had a very shiny nose (like a light bulb)
And if you ever saw it (saw it)
You would even say it glowed (like a flashlight)
All of the other reindeer (reindeer)
Used to laugh and call him names (like Pinocchio)
They never let poor Rudolph (Rudolph)
Join in any reindeer games (like Monopoly)

Then one foggy Christmas Eve, Santa came to say (In his underwear! OR Ho, ho, ho)
Rudolph with your nose so bright
Won't you guide my sleigh tonight

Then how the reindeer loved him (loved him)
And they shouted out with glee (yippee! OR in one town in rural NY they said "like toothpaste")
Rudolph the red-nosed reindeer (reindeer)
You'll go down in history...(like George Washington)