This paper introduces an extension of the vehicle routing problem by including several distributors in competition. Each customer is characterized by demand and a wholesale price. Under this scenario a solution may have unserviced customers and elementary routes with no customer visits. The problem is described as a vehicle routing game (VRG) with coordinated strategies. We provide a computable procedure to calculate a strong equilibrium (SE) in the VRG that is stable against deviations from any coalition. Following this procedure, we solve iteratively optimization subproblems for a single distributor, reducing the set of unserviced customers at each iteration. We prove that strong equilibria of one type exist for a VRG, and we provide conditions for another type to exist. We also introduce a semi-cooperative SE that helps reduce a set of strong equilibria in the VRG. Our methodology is suited for parallel computing, and could be efficiently applied to routing vehicles with a few compartments. It also calcula