The classical Oka-Weil theorem states that any holomorphic function defined on a neighborhood of a polynomially convex compact set can be uniformly approximated by polynomials. We extend this result to approximation by certain subrings of polynomials, specifically those whose exponents lie in the cone spanned by a given compact convex set. Our motivation for studying these polynomial classes arises from the development of a corresponding version of pluripotential theory, which is of independent interest. In particular, we establish a Siciak-Zakharyuta-type theorem in this setting.