Azizi is correct that "zip" = speedy first appeared in print in 1852: in the Knickerbocker, vol. XL, p. 182; "How did we z-i-p! Seven miles at one time, in less than seven minutes." OEL, 1987 Supplement.

When the term was first used, of course, is open to question.

Zip Coon seems to have been sung first by Thomas Birch at Atwill's Music Saloon, New York, 1834.

1st verse-
O ole Zip Coon he is a larned skoler,
O ole Zip Coon he is a larned skoler,
O ole Zip Coon he is a larned skoler,
*Sings posum up a gum tree an coony in a holler,
Posum up a gum tree, coony on a stump,
Posum up a gum tree, coony on a stump,
Den over dubble trubble, Zip Coon will jump.

* reference to another song of the time.

(Chorus as Azizi has posted).

"Zip" is used as a name; nothing in the song equates it with speed. Whether it is a name or nickname (Scipio) or was just used as a handy name by the composer - ?

I believe this American music hall song already is posted, but too lazy to look it up.