We introduce a cohomology theory for finite group schemes with commutative formal groups as coefficients. Using Fontaine's Witt covectors, this theory provides a p-adic cohomology theory for finite group schemes and is motivated by the failure of étale cohomology to detect inseparable extensions. We define a G-module structure on commutative formal group schemes and prove that their category forms a Grothendieck category, so it has enough injectives. We show that, with Witt covectors as coefficients, the derived functors of the invariants functor coincide with the cohomology computed via the bar resolution. As an application, we identify the first cohomology of a finite commutative p-group scheme G with its Dieudonné module.