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This work presents a distributed method for multi-vehicle coordination based on nonlinear model predictive control (NMPC) and dual decomposition. Our approach allows the vehicles to coordinate in tight spaces (e.g., busy highway lanes or parking lots) by using a polytopic description of each vehicle's shape and formulating collision avoidance as a dual optimization problem. Our method accommodates heterogeneous teams of vehicles (i.e., vehicles with different polytopic shapes and dynamic models can be part of the same team). Our method allows the vehicles to share their intentions in a distributed fashion without relying on a central coordinator and efficiently provides collision-free trajectories for the vehicles. In addition, our method decouples the individual-vehicles' trajectory optimization from their collision-avoidance objectives enhancing the scalability of the method and allowing one to exploit parallel hardware architectures. All these features are particularly important for vehicular applications, where the systems operate at high-frequency rates in dynamic environments. To validate our method, we apply it in a vehicular application, that is, the autonomous lane-merging of a team of connected vehicles to form a platoon. We compare our design with the centralized NMPC design to show the computational benefits of the proposed distributed algorithm.