No. The category of commutative Hopf algebras over a field is opposite to the category of group schemes, so your question is "is every epi of $k$-group schemes surjective." But there are maps of finite groups that are epis, but not surjective (this happens when the image has normal closure the whole group), for example the inclusion of a transposition into $S_3$.

I suspect every epi in commutative Hopf algebras is surjective. Certainly one can't construct an example from finite groups. I might be forgetting some funniness about group schemes.

No. The category of commutative Hopf algebras over a field is opposite to the category of affine group schemes, so your question is "is every epi of affine $k$-group schemes surjective?" But there are maps of finite groups that are epis, but not surjective (this happens when the image has normal closure the whole group), for example the inclusion of a transposition into $S_3$.

I suspect every epi in commutative Hopf algebras is surjective. Certainly one can't construct an example from finite groups. I might be forgetting some funniness about group schemes.