We present a virtual element method for a problem with a moving elastic interface in Stokes flow. On a background Cartesian mesh of the domain, each element not cut by the interface consists of eight nodes, namely, the four vertices and the mid-points of four edges. For each cut-element, we move the mid-node onto the interface location, which not only updates the element connectivity, but also allows for flexibly matching the interface as time advances. This simple and effective idea inspires us to develop an interface-fitted mesh generator. In spatial discretization, a linear virtual element approximation is developed for the Stokes problem, which delivers a velocity with local mass conservation. Then, a semi-implicit discretization is designed for the Stokes equation with immersed moving interface, where the discrete bilinear forms are concise without the use of both additional penalty terms and multipliers.Theoretically, we prove that the discrete scheme is unconditionally stable. Finally, the efficiency and accuracy of our method are verified by extensive numerical examples, including the optimal convergence rates in appropriate norms, the capacity to accurately track the interface evolution. \begin{center} \fontsize{11}{12}\selectfont \begin{thebibliography}{99} \bibitem{Paper} L. Beir\~{a}o da Veiga, K. Lipnikov, \emph{A mimetic discretization of the {S}tokes problem with selected edge bubbles}, SIAM J. Sci. Comput., 32, 875-893, 2010. \bibitem{Paper} G. Manzini, A. Mazzia, \emph{A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem}, J. Comput. Dyn., 9, 207-238, 2022. \bibitem{Paper} G. Manzini, A. Mazzia, \emph{Conforming virtual element approximations of the two-dimensional Stokes problem}, Appl. Numer. Math., 181, 176-203, 2022. \bibitem{Paper} G. Laymuns, Manuel A. S\'{a}nchez, \emph{Corrected finite element methods on unfitted meshes for Stokes moving interface problem}, Comput. Math. Appl., 108, 159-174, 2022. \bibitem{Paper} Y. Mori, A. Rodenberg, D.Spirn, \emph{Well-posedness and global behavior of the Peskin problem of an immersed elastic filament in Stokes flow}, Comm. Pure Appl. Math., 72, 887-980, 2019. \end{thebibliography} \end{center}