The exponential increase in orbital debris and active satellites will lead to congested orbits, necessitating more frequent collision avoidance maneuvers by satellites. To minimize fuel consumption while ensuring the safety of satellites, enforcing a chance constraint, which poses an upper bound in collision probability with debris, can serve as an intuitive safety measure. However, accurately evaluating collision probability, which is critical for the effective implementation of chance constraints, remains a non-trivial task. This difficulty arises because uncertainty propagation in nonlinear orbit dynamics typically provides only limited information, such as finite samples or moment estimates about the underlying arbitrary non-Gaussian distributions. Furthermore, even if the full distribution were known, it remains unclear how to effectively compute chance constraints with such non-Gaussian distributions. To address these challenges, we propose a distributionally robust chance-constrained collision avoidance algorithm that provides a sufficient condition for collision probabilities under limited information about the underlying non-Gaussian distribution. Our distributionally robust approach satisfies the chance constraint for all debris position distributions sharing a given mean and covariance, thereby enabling the enforcement of chance constraints with limited distributional information. To achieve computational tractability, the chance constraint is approximated using a Conditional Value-at-Risk (CVaR) constraint, which gives a conservative and tractable approximation of the distributionally robust chance constraint. We validate our algorithm on a real-world inspired satellite-debris conjunction scenario with different uncertainty propagation methods and show that our controller can effectively avoid collisions.