Artificial atom qubits in diamond have emerged as leading candidates for a range of solid-state quantum systems, from quantum sensors to repeater nodes in memory-enhanced quantum communication. Inversion-symmetric group IV vacancy centers, comprised of Si, Ge, Sn and Pb dopants, hold particular promise as their neutrally charged electronic configuration results in a ground-state spin triplet, enabling long spin coherence above cryogenic temperatures. However, despite the tremendous interest in these defects, a theoretical understanding of the electronic and spin structure of these centers remains elusive. In this context, we predict the ground- and excited-state properties of the neutral group IV color centers from first principles. We capture the product Jahn-Teller effect found in the excited state manifold to second order in electron-phonon coupling, and present a non-perturbative treatment of the effect of spin-orbit coupling. Importantly, we find that spin-orbit splitting is strongly quenched due to the dominant Jahn-Teller effect, with the lowest optically-active ³E_u state weakly split into m_s-resolved states. The predicted complex vibronic spectra of the neutral group IV color centers are essential for their experimental identification and have key implications for use of these systems in quantum information science.