The electronic structure of semiconductors and insulators is affected by ionic motion through electron-phonon interaction, yielding temperature-dependent band gap energies and zero-point renormalization (ZPR) at absolute zero temperature. For polar materials, the most significant contribution to the band gap ZPR can be understood in terms of the Fröhlich model, which focuses on the non-adiabatic interaction between an electron and the macroscopic electrical polarization created by a long-wavelength optical longitudinal phonon mode. On the other hand, spin-orbit interaction (SOC) modifies the bare electronic structure, which will, in turn, affect the electron-phonon interaction and the ZPR. We present a comparative investigation of the effect of SOC on the band gap ZPR of twenty semiconductors and insulators with cubic symmetry using first-principles calculations. We observe a SOC-induced decrease of the ZPR, up to 30%, driven by the valence band edge, which almost entirely originates from the modification of the bare electronic eigenenergies and the decrease of the hole effective masses near the Γ point. We also incorporate SOC into a generalized Fröhlich model, addressing the Dresselhaus splitting which occurs in non-centrosymmetric materials, and confirm that the predominance of non-adiabatic effects on the band gap ZPR of polar materials is unchanged when including SOC. Our generalized Fröhlich model with SOC provides a reliable estimate of the SOC-induced decrease of the polaron formation energy obtained from first principles and brings to light some fundamental subtleties in the numerical evaluation of the effective masses with SOC for non-centrosymmetric materials. We finally warn about a possible breakdown of the parabolic approximation, one of the most fundamental assumptions of the Fröhlich model, within the physically relevant energy range of the Fröhlich interaction for materials with high phonon frequencies treated with SOC. This is a post-peer-review version of the article published in Physical Review B, which includes the Supplemental Material in the main file.