It has been proposed that feebly-interacting ultralight scalars may constitute the dark matter content of the universe. Models describing the interactions of a dark matter scalar with Standard Model fields may feature quadratic interactions at leading order, such that the scalar acquires an effective mass in the neighbourhood of a classical matter distribution. The effect of their introduction is to provide effective shifts in fundamental constants of physics, as well as to mediate scalar fifth forces between test bodies. We here demonstrate how these fifth forces can result in pericentre precession in the orbital motion of satellites around the Earth. We apply this to the measured pericentre precession of the LAGEOS II experiment, constraining the mass, and couplings to the light Standard Model fields, of a quadratically coupled ultra-light dark matter scalar. We observe such an experiment to be effective in constraining parameter space at strong couplings, where existing constraints from satellite and tabletop level experiments break down.