The emerging field of cavity quantum materials seeks to control electronic phases using the quantum fluctuations of confined electromagnetic fields. Most existing approaches rely on electric-dipole, density-like couplings, for which strong equilibrium effects in solids remain difficult to obtain. Here we propose a different route, in which the ground state of an electron system is controlled or modified via coupling to the vacuum fluctuations of the quantized magnetic flux of a superconducting LC circuit. The resulting long-range attractive interactions between electron angular momentum states induce unconventional chiral superconductivity in a broad class of materials, with critical temperatures of a few Kelvin or even higher, depending on the area covered by the field. The magnetic flux of an LC circuit remains in the ground state, and all of its parameters can be controlled to optimize a particular pairing regime. The results identify circuit quantum magnetostatics as a highly tunable platform for engineering electron interactions and creating new quantum phases of matter.