I searched Google Books for the words "Down by the station, early in the morning, see the little * all in a row"—using an asterisk "wild card" to substitute for the questionable term.

I found lots of current books of children's songs that include those lines. They have various ways of spelling—

puffa-bellies (once)
puffabillies (once)
puff-a-billies (once)
puffer bellies (5 occurrences)
puffer billies (3 occurrences)
pufferbellies (3 occurrences)
puffer-bellies (3 occurrences)
pufferbillies (2 occurrences)
puffer-billies (once)
puffinbellies (once)

If you ignore the hyphens and spaces, you get—

puffabellies (once)
puffabillies (2 occurrences)
puffabillies (once)
pufferbellies (11 occurrences)
pufferbillies (6 occurrences)
puffinbellies (once)

Anyone interested in seeing a pie chart? ;-)

But the oldest version I found was in this magazine or journal:

Recreation by National Recreation Association, Vol. 25, 1931, Page 375:

PUFFERBILLY SONG

Down at the station, early in the morning,
See the little Pufferbillies standing in a row.
See the engine driver twist the little handle,
Tst!—Tst!—Toot!—Toot! There they go.

(May be used as a two or four part round. Music to this found in the Girl Scout Song Book.)