We investigate electron paring in a two-dimensional electron system mediated by vacuum fluctuations inside a nanoplasmonic terahertz cavity. We show that the structured cavity vacuum can induce long-range attractive interactions between current fluctuations which lead to pairing in generic materials with critical temperatures in the low-kelvin regime for realistic parameters. The induced state is a pair-density wave superconductor which can show a transition from a fully gapped to a partially gapped phase - akin to the pseudogap phase in high-Tc superconductors. Our findings provide a promising tool for engineering intrinsic electron interactions in two-dimensional materials.