For most CI problems, it is not practical to build and store the Hamiltonian matrix. Rather, one typically employs so-called direct CI schemes that employ an iterative eigensolver for one or a few states where the requisite Hamiltonian matrix elements are computed on the fly. Here, we outline the adaptation of such a direct approach to the QED-CASCI problem specifically for the Hamiltonian in the coherent state basis (Eq. 5); as noted in Equations 10-14, one can follow the details with a few key substitutions in the integrals to obtain the implementation for the photon number basis (Eq. 1).