In this paper, we propose a path planning method for nonholonomic multi-vehicle system in presence of moving obstacles. The objective is to find multiple fixed length paths for multiple vehicles with the following properties: (i) bounded curvature (ii) obstacle avoidant (iii) collision free. Our approach is based on polygonal approximation of a continuous curve. Using this idea, we formulate an arbitrarily fine relaxation of the path planning problem as a nonconvex feasibility optimization problem. Then, we propound a nonsmooth dynamical systems approach to find feasible solutions of this optimization problem. It is shown that the trajectories of the nonsmooth dynamical system always converge to some equilibria that correspond to the set of feasible solutions of the relaxed problem. The proposed framework can handle more complex mission scenarios for multi-vehicle systems such as rendezvous and area coverage.