This paper examines a stochastic formulation of the generalized Nash equilibrium problem where agents are subject to randomness in the environment of unknown statistical distribution. We focus on fully distributed online learning by agents and employ penalized individual cost functions to deal with coupled constraints. Three stochastic gradient strategies are developed with constant step-sizes. We allow the agents to use heterogeneous step-sizes and show that the penalty solution is able to approach the Nash equilibrium in a stable manner within O(μmax), for small step-size value μmax and sufficiently large penalty parameters. The operation of the algorithm is illustrated by considering the network Cournot competition problem.