Silicon carbide with optically and magnetically active point defects offers unique opportunities for quantum technology applications. Since interaction with these defects commonly happens through optical excitation and de-excitation, a complete understanding of their light-matter interaction in general and optical signatures, in particular, is crucial. Here, we employ quantum mechanical density functional theory calculations to investigate the photoluminescence lineshapes of selected, experimentally observed color centers (including single vacancies, double vacancies, and vacancy impurity pairs) in 4H-SiC. The analysis of zero-phonon lines as well as Huang-Rhys and Debye-Waller factors are accompanied by a detailed study of the underlying lattice vibrations. We show that the defect lineshapes are governed by strong coupling to bulk phonons at lower energies and localized vibrational modes at higher energies. Generally, good agreement to the available experimental data is obtained, and thus we expect our theoretical work to be beneficial for the identification of defect signatures in the photoluminescence spectra and thereby advance the research in quantum photonics and quantum information processing.