Contact mechanics problems are notoriously challenging in computational mechanics due to the need for accurate contact detection, robust enforcement of interface conditions, and the frequent presence of complex or non-conforming geometries. Embedded and unfitted methods offer an attractive alternative to classical body-fitted approaches by decoupling the computational mesh from the geometric description. Among these, the Shifted Boundary Method (SBM) has recently emerged as an effective technique for the treatment of boundary and interface conditions without resorting to cut-cell integration [1]. In this contribution, we present a finite element formulation of frictionless contact problems based on the Shifted Boundary Method [2]. Contact constraints are enforced on surrogate boundaries, allowing for a consistent and robust treatment of contact interactions while preserving the simplicity of standard finite element discretizations. The proposed approach avoids geometric complexity and remeshing, making it particularly suitable for problems involving complex or imperfect geometrical descriptions. A comprehensive set of numerical benchmarks is presented to assess accuracy, robustness, and convergence properties, including comparisons with established finite element contact formulations based on augmented Lagrangian techniques. Building upon this finite element framework, we further explore an extension of the SBM contact formulation to high-order discretizations using B-spline basis functions, inspired by recent developments of the SBM in spline-based settings [3]. This extension exploits the enhanced smoothness and approximation properties of high-order spline spaces, showing improved accuracy and convergence behavior in representative numerical examples.