Quantum emitters in diamond are leading optically accessible solid-state qubits. Among these, Group IV-vacancy defect centers have attracted great interest as coherent and stable optical interfaces to long-lived spin states. Theory indicates that their inversion symmetry provides first-order insensitivity to stray electric fields, a common limitation for optical coherence in any host material. Here we experimentally quantify this electric field dependence via an external electric field applied to individual tin-vacancy (SnV) centers in diamond. These measurements reveal that the permanent electric dipole moment and polarizability are at least 4 orders of magnitude smaller than for the diamond nitrogen vacancy (NV) centers, representing the first direct measurement of the inversion symmetry protection of a Group IV defect in diamond. Moreover, we show that by modulating the electric-field-induced dipole we can use the SnV as a nanoscale probe of local electric field noise, and we employ this technique to highlight the effect of spectral diffusion on the SnV.