Here's how I hear it:

No hat on his head, no shoes on his feet
You oughta seen them women curling in their flats

My first thought was that flats could also be a type of shoe -- as opposed to high heels.

A second guess is that they're curling with excitement/anticipation/laughter in their apartments (as you suggested) at the idea of a man walking in public in his underwear. But I don't like this line of thought because this is a western song, where most people wouldn't be thinking of apartment flats. However, the original was by Hezekiah Jenkins, an obscure singer who seems to have traveled predominantly around the south, southwest, and midwest, but did have a stint in NYC, especially around the time the song came out. He was best known as a minstrel performer (link).

The third, most likely guess (in my opinion, at least) that I thought of is that it's describing women using curlers to get their hair to curl in an attempt to look nice, either an attempt to to woo the garbless man or look nice for their husbands who aren't yet home.