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In this paper, we study the problem of distributed generalized stochastic Nash equilibrium seeking for robot systems over a connected undirected graph. We use the cost functions containing uncertainty to represent the system’s performance under varying conditions. To mitigate the challenges posed by this uncertainty, we employ the Tikhonov regularization scheme, which also helps us to relax the strongly monotone condition of the cost functions to the strictly monotone condition. We also consider the inequality constraints, which represent the feasible working space of robots. Additionally, auxiliary parameters are introduced in the control laws to facilitate seeing the variational generalized stochastic Nash equilibrium. The convergence of the proposed control laws is analyzed by using the operator splitting method. Finally, we demonstrate the effectiveness of the proposed algorithm through an example involving multiple robots connected through a communication network.