In this paper, the notion of the vector quasiconcavity and lower vector continuity for multivalued mappings without using the algebraic structure are introduced. By applying these definitions and maximal element lemma, some existence theorems of the solution of the system of vector quasi-equilibrium problems for a family of multivalued mappings in the setting of topological order spaces are established. The results of this note improve and generalize the corresponding results in the literature, specially references \cite{AFS,BP,FZ,Fu,FW,AB1}.