The past decades have seen numerous efforts to apply the finite volume methodology to solid mechanics problems. However, only limited work has been done by the finite volume community toward the simulation of mechanical contact. In this article, we present a novel semi-implicit methodology for the solution of static force-loading contact problems with cell-centered finite volume codes. Starting from the similarities with multi-material problems, we derive an implicit discretization scheme for the normal contact stress with a straightforward inclusion of frictional forces and correction vectors for non-orthogonal boundaries. With the introduction of a sigmoid blending function interpolating between contact stresses and gap pressure, the proposed approach is extended to cases with partially closed gap. The contact procedure is designed around an arbitrary mesh mapping algorithm to allow for non-conformal meshes at the contact interface between the two bodies. Finally, we verify the contact methodology against five benchmarks cases.