This work examines a stochastic formulation of the generalized Nash equilibrium problem (GNEP) where agents are subject to randomness in the environment of unknown statistical distribution. Three stochastic gradient strategies are developed by relying on a penalty-based approach where the constrained GNEP formulation is replaced by a penalized unconstrained formulation. It is shown that this penalty solution is able to approach the Nash equilibrium in a stable manner within O(p), for small step-size values p. The operation of the algorithms is illustrated by considering the Cournot competition problem.