This paper studies an n-player non-cooperative game with strategy sets defined by stochastic linear constraints. The stochastic constraints of each player are jointly satisfied with a probability exceeding a given threshold. We consider the case where the row vectors defining the constraints are independent random vectors whose probability distributions are not completely known and belong to a certain distributional uncertainty set. The random constraints of each player are formulated as a distributionally robust joint chance constraint. We consider one density based uncertainty set and four two-moments based uncertainty sets. One of the considered uncertainty set is based on a nonnegative support. Under standard assumptions on players' payoff functions, we show that there exists a Nash equilibrium of a distributionally robust chance-constrained game for each uncertainty set.