In this paper, we consider symmetric weak vector quasi-equilibrium problems and introduce the concepts of Levitin-Polyak well-posedness and Levitin-Polyak well-posedness in the generalized sense for these problems. Then, we show that, under suitable conditions, the equivalence between the Levitin-Polyak well-posedness propertises and the existence of solutions for the considered problems is given. Further, some metric characterizations of the Levitin-Polyak well-posedness and Levitin-Polyak well-posedness in the genealized sense for such problems in terms of the behavior of the approximate solution sets are also discussed. Finally, as an application, we study the special case of symmetric weak vector quasi-variational inequality problems. The results presented in this paper improve and extend some main results in the liteature. Some examples are given for the illustration of our results.