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 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. Specifically, we will focus on the displacement-pressure formulation of a frictionless contact problem between two elastic bodies in the nearly incompressible regime. After a brief introduction to the mathematical model of the contact problem and to its well-posedness, we will present an explicit construction of the corresponding VEM discretization. Next, we will analyze the convergence result of the proposed method, showing its robustness with respect to the incompressibility parameter. Some numerical experiments will be presented to validate theoretical estimates. Finally, we will emphasize the main advantages of VEMs in the discrete enforcement of contact conditions, focusing in particular on the node insertion strategy and the management of small edges.