We propose an approach to the problem of computing a minimum-time tour through a series of waypoints for a Dubins vehicle in the presence of stochasticity. In this paper, we explicitly account for kinematic nonlinearities, the stochastic drift of the vehicle, the stochastic motion of the targets, and the possibility for the vehicle to service each of the targets or waypoints, leading to a new version of the Dubins vehicle traveling salesperson problem (TSP). Based on the Hamilton–Jacobi–Bellman (HJB) equation, we first compute the minimum expected time feedback control to reach one waypoint. Next, minimum expected times associated with the feedback control are used to construct and solve a TSP. We provide numerical results illustrating our solution, analyze how the stochasticity affects the solution, and consider the possibility for on-line recomputation of the waypoint ordering in a receding-horizon manner.