Are supervector spaces the representations of a Hopf algebra?

As a monoidal category, supervector spaces are the same as $mathbb{Z}/2mathbb{Z}$ representations, so you can think of them as representations of the Hopf algebra $mathbb{C}[mathbb{Z}/2mathbb{Z}]$.

However, the symmetric structure is different, so the thing you need to do is make $mathbb{C}[mathbb{Z}/2mathbb{Z}]$ into a quasi-triangular Hopf algebra in an interesting way, that is, change things so that the map from $Votimes Wto Wotimes V$ isn't just flipping, it's flipping followed by an element of $mathbb{C}[mathbb{Z}/2mathbb{Z}]otimes mathbb{C}[mathbb{Z}/2mathbb{Z}]$ which is called the R-matrix. I'll leave actually writing this element as an exercise to the reader, but it's uniquely determined by acting by -1 on sign tensor sign and by 1 on everything else.

You may want to have a look at the relevant Wikipedia article.