We consider a two player zero-sum game with stochastic linear constraints. The probability distributions of the vectors associated with the constraints are partially known. The available information with respect to the distribution is based mainly on the two first moments. In this vein, we formulate the stochastic linear constraints as distributionally robust chance constraints. We consider three different types of moments based uncertainty sets. For each uncertainty set, we show that there exists a saddle point equilibrium of the game. The latter corresponds to the optimal solution of a primal-dual pair of second order cone programs. In order to illustrate our theoretical results, we consider an instance of a zero-sum game together with distributionally robust chance constraints. A comparison with expected value constraints zero-sum game is drawn to illustrate the effectiveness of our approach.