Developing accurate and computationally efficient models for underwater sound propagation in the uncertain, dynamic ocean environment is inherently challenging. In this work, we evaluate the potential of dynamic reduced-order modeling for stochastic ray tracing. We obtain and implement the stochastic dynamically-orthogonal (DO) differential equations for Ray Tracing (DO-Ray). With stochastic DO-Ray, we can start from non-Gaussian environmental uncertainties and compute the stochastic acoustic ray fields in a dynamic reduced order fashion, all while preserving the dominant complex statistics of the ocean environment and the nonlinear relations with ray dynamics. We develop varied algorithms and discuss implementation challenges and solutions, using direct Monte Carlo for comparison. We showcase results in an uncertain deep-sound channel example and observe the ability to represent the stochastic ray trace fields in a dynamic reduced-order fashion.