In this paper, a game-theoretic approach is considered for the vehicle routing problem with many distributors. Each customer is characterized by demand and wholesale price. Within such a statement, some customers are possibly not visited by a distributor in the optimal solution. This problem is called the vehicle routing game (VRG) in coordinated strategies. A procedure for determining a strong equilibrium in the VRG is proposed which is stable against coalitional deviations. According to the procedure, the optimization problem is solved iteratively for each distributor. The set of unvisited customers is reduced at each step. The existence of two classes of strong equilibria is proved. The concept of a cooperative strong equilibrium is presented. All results are illustrated by numerical examples.