The observation that the entropy of entanglement for groundstates of quantum mechanical systems often scales like the area of the region separating the two rather than the volume. These weak correlations allow a host of quantum techniques such as matrix product states and DMRG to provide accurate estimates. However, existing area law results fail to rigorously hold for systems that have unbounded operators such as those that naturally appear in lattice gauge theories. Here we rectify this problem by proving an entanglement area law for a class of 1D quantum systems involving infinite-dimensional local Hilbert spaces. This class of quantum systems include bosonic models such as the Hubbard-Holstein model, and both U(1) and SU(2) lattice gauge theories in one spatial dimension. Our proof relies on new results concerning the robustness of the ground state and spectral gap to the truncation of Hilbert space, applied within the approximate ground state projector (AGSP) framework from previous work. In establishing this area law, we develop a system-size independent bound on the expectation value of local observables for Hamiltonians without translation symmetry, which may be of separate interest. Our result provides theoretical justification for using tensor network methods to study the ground state properties of quantum systems with infinite local degrees of freedom.