There are several recordings available. I was able to piece together a few more lyrics from sound samples:

WHO'S TAKING YOU HOME TONIGHT?
Words, Tommy Connor. Music, Manning Sherwin. 1939.

They met at a dance...
...at these words from the boy of her dreams:

Who's taking you home tonight after the dance is through?
Who's going to hold you tight and whisper "I love you, I do"?
Who's the lucky boy that's going your way
To kiss you goodnight at your doorway?
Who's taking you home tonight? Darling, it's plain to see,
I'm pleading, please let it be me....

[As sung by Vera Lynn. There are also recordings by Maurice Winnick and His Orchestra with vocalist Al Bowlly, Oscar Rabin, and Sydney Lipton. (The latter 2 may be instrumentals.)]

Sorry I took so long to respond. I would have noticed this thread sooner if it had had the prefix "Lyr Req:".