Virtual Element Methods (in short, VEMs) are a recent family of numerical methods widely employed today for approximating partial differential equations. This class of Galerkin methods naturally adapts to arbitrary polygonal decompositions of the domain, due to the choice of suitable discrete spaces that are no longer restricted to polynomials. This flexibility makes VEMs particularly well-suited for dealing with variational problems with complex geometries and non-standard boundary conditions. In this talk, we will explore the application of the Stokes-like virtual element method to address a fundamental problem in solid mechanics, known as the contact problem. After a brief introduction to the mathematical framework of VEMs, we will focus on the displacement-pressure formulation of a frictionless contact problem between two elastic bodies, specifically in the nearly incompressible regime. We will discuss results related to the existence and uniqueness of the solution for the continuous problem. Furthermore, we will present an explicit construction of the VEM discretization for this problem, along with the corresponding convergence results. Finally, we will highlight the key advantages of VEMs in the discrete treatment of contact conditions compared to classical FEMs and we will show some numerical tests that validates theoretical estimates.