As with many etymologies, the origins aren't necessarily clear.
Thomas Pinney in the book A History of Wine in America
gives several possible explanations, breaking them into categories
Aesop's fable of the fox and the grapes - he directly dismisses this as being fanciful
"fox" does not refer to the animal, but instead to "wild", "intoxicated" or a corruption of the French word "faux"
"fox" refers to the appearance of the leaves, either because they look like fox paws or because they have fox-colored fuzz.
Animals (specifically foxes) are attracted to them, either because they like to eat them, or are attracted to the smell.
The grape has a musky odor like a fox.
Of all the origins, most are only attested in a small number of sources that Pinney can find, or are of a relatively recent origin (ca. 1900 or later) despite the term "fox grape" being attested back to the early 1600s.
Of the proffered origins, it's only that of the musky odor being like a fox that has multiple sources which date back to the mid 1600s, so it may be that it is indeed the origin of the term "fox grape", even though "foxy", when applied to wine, derives from the name of the grape, rather than the odor directly.
Nothing is certain, though, as Pinney references Liberty Hyde Bailey's Sketch of the Evolution of Our Native Fruits which posits that the term "fox grape" could ultimately be from another source (e.g. "intoxicated"), which was then folk etymologied into "smelling like a fox" at a slightly later date.