Color centers in wide-band-gap semiconductors are leading platforms for solid-state quantum technologies, yet a quantitative description of their vibronic structure has remained elusive due to the complexity of multi-phonon processes in localized defect states. Here we present a first-principles Green’s function framework based on the retarded cumulant ansatz (RCA) to describe electron-phonon interactions in spin defects; our approach goes beyond the adiabatic and lowest-order perturbation theory approximations underlying widely used approaches. Applied to the negatively charged nitrogen-vacancy (NV${}^{-}$) center in diamond within the harmonic approximation, our method reveals that multi-phonon satellites persist over a 400 meV energy window even at zero temperature, driven by quantum zero-point fluctuations. We demonstrate that accurate spectral functions require mode-, momentum-, spin-, and orbital-resolved electron-phonon matrix elements sampled across the full Brillouin zone, to account for hybridized and propagating phonon channels. We find that the vibronic structure of the NV− center exhibits strong spin and orbital anisotropy, with different orbitals coupling to qualitatively distinct parts of the phonon spectrum, and spin-selective coupling affecting both sideband positions and intensities.