This paper focuses on distributionally robust controller design for avoiding dynamic and stochastic obstacles whose exact probability distribution is unknown. The true probability distribution of the disturbance associated with an obstacle, although unknown, is considered to belong to an ambiguity set that includes all the probability distributions that share the same first two moment. The controller thus focuses on ensuring the satisfaction of the probabilistic collision avoidance constraints for all probability distributions in the ambiguity set, hence making the solution robust to the true probability distribution of the stochastic obstacles. Techniques from robust optimization methods are used to model the distributionally robust probabilistic or chance constraints as a semi-definite programming (SDP) problem with linear matrix inequality (LMI) constraints that can be solved in a computationally tractable fashion. Simulation results for a robot obstacle avoidance problem shows the efficacy of our method.