This work introduces and analyzes Proximal Galerkin (PG) schemes for solving variational problems with pointwise inequality constraints in vector-valued function spaces such as H(div; Ω) ∩ H(curl; Ω). Natural discretizations for unconstrained problems of this type yield a saddle-point structure via the Finite Element Exterior Calculus (FEEC). This extends to a sequence of twofold saddle-point sub-problems when applying the PG methodology to the constrained analogue. Previous work applying PG to constrained minimization of second-order, elliptic energy functionals demonstrated the benefits of twofold saddle-point problems in anisotropic diffusion and obstacle problems, including favorable context-specific properties such as local mass conservation. Here, we expand the framework across the de Rham complex, establishing the well-posedness of the general twofold saddle-point systems solved at each proximal iteration. Furthermore, we extend the analysis of PG schemes by providing a priori error analysis for inequality constraints in H(div; Ω) and H(curl; Ω), while past work only concerned bound constraints in the H1(Ω) setting. We demonstrate our approach by showcasing one of the first numerical methods for vector Laplacian obstacle problems associated with the 2D and 3D de Rham complexes. While discretizations of vector Laplacian problems have been studied extensively through FEEC, the constrained variants have not previously undergone a systematic analysis. Possible applications include modeling constrained electromagnetism. Finally, this framework yields a new approach to the numerical solution of the Hintermüller–Kunisch formulation of the BV-regularized image restoration problem.