Mitford Mathews, A Dictionary of Americanisms on Historical Principles (1951), offers this entry for the word snapper:
snapper, n. 1. = snapping turtle. [Citations from 1796 forward omitted.] ... 2. A cracker at the end of a whip. Usu. transf. in the sense of a word or phrase giving a smart or pointed finish to something. [First three cited examples:] 1835 Hoffman, Winter in West I. 179 Jim cracked his snapper. 1857 Holland Bay Path xiv, You'd a said twenty lashes ... and Mr. Moxon would 'a said twenty Amens on the end on 'em for a snapper. 1895 Mark Twain How to Tell a Story 226 (R.) A humorous story finishes with a nub, point, snapper, or whatever you like to call it.
In definition 2, the "cracker" refers (I believe) to a bit of thong attached to the end of the whip to give it (when used properly) an extra-loud cracking sound. I have seen and used bullwhips that have this extra piece of thong on them, but I haven't heard them called "crackers" (or "snappers").
The citation from 1835 uses snapper in its literal sense, and could very well be taken to refer to the entire whip; the two later citations use snapper figuratively.
The word appears in its literal sense in Philip Paxton, A Stray Yankee in Texas (1853), in the context of a hostile interaction between the Yankee (Green) and a mule (Brandy):
GREEN. There, take that (attempting an application of the whip, and only succeeding in getting a smart rap with the snapper upon his cheek). Rot these darn fool whips!—as long as the moral law'n the ten commandments with the hull book a Revelation for a snapper.
After various attempts, Green began, as he said, "to get the hang of the thing," and then commenced a race around the lot, the Yankee cracking away at the mule and getting rather the larger share of the lash himself, until he finally cornered his antagonist in a kind of cul-de-sac, formed by the junction of the fence and stable at a very acute angle.
Evidently, in the sentence "There's a woman with a snapper," the snapper is a whip, which makes sense given that the woman in question (Mrs. Morewood) is described as "driving a pair of horses."