We propose an active-set algorithm for smooth multiobjective optimization problems subject to box constraints. The method works on one face of the feasible set at a time, treating it as a lower-dimensional region on which the problem simplifies. At each iteration, the algorithm decides whether to remain on the current face or to move to a different one, characterizing two types of iterations: face-exploring and face-abandoning steps. Backtracking and extrapolation strategies are combined, allowing the working set to be expanded or reduced by multiple constraints in a single iteration. Global convergence to Pareto critical points is established, and under a dual-nondegeneracy assumption we prove finite identification of the active set. Implementation aspects are discussed in detail, and numerical experiments on benchmark problems illustrate the practical performance of the method in comparison with existing approaches.