Quantum localization via atomic point defects in semiconductors is of significant fundamental and technological importance. Quantum defects in monolayer transition-metal dichalcogenide semiconductors have been proposed as stable and scalable optically-addressable spin qubits. Yet, the impact of strong spin-orbit coupling on their dynamical response, for example under optical excitation, has remained elusive. In this context, we study the effect of spin-orbit coupling on the electron-phonon interaction in a single chalcogen vacancy defect in monolayer transition metal dichalcogenides, molybdenum disulfide (MoS2) and tungsten disulfide (WS2). From ab initio electronic structure theory calculations, we find that spin-orbit interactions tune the magnitude of the electron-phonon coupling in both optical and charge-state transitions of the defect, modulating their respective efficiencies. This observation opens up a promising scheme of dynamically modulating material properties to tune the local behavior of a quantum defect.