We present dielectric, polarization, resistivity, specific-heat, and magnetic susceptibility data on single crystals of the lacunar spinels GaNb4Se8 and GaTa4Se8, tetrahedral cluster-based materials with substantial spin-orbit coupling. We concentrate on the possible occurrence of antipolar order in these compounds, as previously reported for the isoelectronic GaNb4S8, where spin-orbit coupling plays a less important role. Our broadband dielectric-spectroscopy investigations reveal clear anomalies of the intrinsic dielectric constant at the magnetostructural transitions in both systems that are in accord with the expectations for antipolar transitions. A similar anomaly is also observed at the cubic-cubic transition of the Nb compound leading to an intermediate phase. Similar to other polar and antipolar lacunar spinels, we find indications for dipolar relaxation dynamics at low temperatures. Polarization measurements on GaNb4Se8 reveal weak ferroelectric ordering below the magnetostructural transition, either superimposed to antipolar order or emerging at structural domain walls. The temperature-dependent dc resistivity evidences essentially thermally activated charge transport with different activation energies in the different phases. A huge steplike increase of the resistivity at the magnetostructural transition of the Ta compound points to a fundamental change in the electronic structure or the mechanism of the charge transport. At low temperatures, charge transport is governed by in-gap impurity states, as also invoked to explain the resistive switching in these compounds.