Point defects in semiconductors are promising single-photon emitters (SPEs) for quantum computing, communication and sensing applications. However, factors such as emission brightness, purity and indistinguishability are limited by interactions between localized defect states and the surrounding environment. Therefore, it is important to map the full emission spectrum from each SPE, to understand the complex interplay between the different defect configurations, their surroundings and external perturbations. Herein, we investigate a family of regularly spaced sharp luminescence peaks appearing in the near-infrared portion of photoluminescence (PL) spectra from n-type 4H-SiC samples after irradiation. This periodic emitter family, labeled the L-lines, is only observed when the zero-phonon line signatures of the negatively charged Si vacancy (so-called V-lines) are present. The L-lines appear with 1.45 meV and 1.59 meV energy spacing after H and He irradiation and increase linearly in intensity with fluence -reminiscent of the intrinsic defect trend. Furthermore, we monitor the dependence of the L-line emission energy and intensity on heat treatments, electric field strength and PL collection temperature, discussing these data in the context of the L-lines. Based on the strong similarity between the irradiation, electric field and thermal responses of the L-and V-lines, the L-lines are attributed to the Si vacancy in 4H-SiC. The regular and periodic appearance of the L-lines provides strong arguments for a vibronic origin explaining the oscillatory multi-peak spectrum. To account for the small energy separation of the L-lines, we propose a model based on rotations of distortion surrounding the Si vacancy driven by a dynamic Jahn-Teller effect.