In this paper, we consider strong vector equilibrium problems in normed spaces. Firstly, we study stability conditions for a scalar equilibrium problem without assuming boundedness and concavity properties of the constraint map and objective function, respectively. Next, we discuss properties of the Hiriart-Urruty oriented distance function in an ordered space, and then using these properties, relationships between the strong equilibrium problems and the scalar ones are formulated. Then after, based on these relationships, we address sufficient conditions for the Lipschitz continuity of approximate solution maps to vector equilibrium problems via the corresponding results of the scalar equilibrium problems. As an application, we apply the obtained results to express stability conditions for network equilibrium problems.