The accurate and efficient simulation of contact problems remains one of the most challenging tasks in computational mechanics, particularly when dealing with complex geometries. The Isogeometric Analysis (IGA) framework offers a robust solution by enabling smooth, high-order, and geometrically exact representations. However, the construction of watertight, analysis-suitable parameterizations remains a significant bottleneck in many practical applications. Unfitted or embedded techniques represent a promising alternative by decoupling the mesh from the geometric model. Among these, the Shifted Boundary Method (SBM) has emerged as a versatile embedded technique that avoids cut-cell integration and simplifies geometry handling. In the IGA context, the SBM further leverages the advantages of high-order continuity and exact geometry representation. In this work we make use of the SBM in the IGA framework for both structural and frictionless contact mechanics problems. We formulated a high-order SBM-IGA discretization for linear elasticity with mixed boundary conditions, which is the base for the developed Nitsche-based, variationally consistent contact formulation over surrogate boundaries, built upon recent advances in unbiased contact enforcement. Numerical results confirm the robustness and accuracy of the method, including comparisons with classical body-fitted IGA and a state-of-the-art Finite Element Method based on the augmented Lagrangian contact algorithm. The proposed SBM-IGA approach demonstrates excellent potential for simulating contact problems involving non-watertight CAD or imaging-based geometries. All simulations are performed using the open-source Kratos Multiphysics framework.