Yes, you can certainly use group VI elements for n-type doping in semiconductors which contain pentavalent atoms. Some examples are: Sulfur, Selenium, and Tellurium which turn Gallium Arsenide into an n-type semiconductor. However, Gallium Arsenide is a compound semiconductor; and I don't think elements in groups II, III, V, VI can form semiconductors by themselves (i.e. without combining with elements from other groups). You can see a list of semiconductors here:
You can see that there are mostly group IV, III-V, and II-VI present.
As Ron said, carbon, in diamond form, would be pretty expensive. However, in 2004 graphene was found to be another allotrope of carbon; it is turning out to be a promising candidate for electronics. The electron mobility is orders of magnitude better than Silicon. As a matter of fact, it is better than the best semiconductors (such as InSb) known so far. Graphene, unlike diamond, is very cheap. You produce some of it every time you write with a pencil! Also, you can dope graphene the same way as silicon; i.e. with group V elements to become an n-type semiconductor. But the interesting thing about graphene is that you can also dope it electrostatically.