The origin of superconductivity in oxide interfaces and its relation to ferroelectricity remains an open question. At LaAlO₃/SrTiO₃ interfaces, quantum confinement and inversion symmetry breaking create a two-dimensional electron gas near a ferroelectric quantum critical point, yet direct evidence linking phonon dynamics to electron pairing has been lacking. Here we directly probe lattice vibrations and atomic structure at LaAlO₃/SrTiO₃ interfaces across the superconducting phase diagram using vibrational spectroscopy with momentum selectivity in a scanning transmission electron microscope. We find that superconductivity across the doping series correlates with inversion symmetry breaking and the appearance of high-frequency localized phonons. These tunable, polar vibrations—confined near the interface—exhibit strong electron–phonon coupling and evolve systematically with carrier density. Our findings establish a link between lattice instability, superconductivity and strong electron–phonon coupling mediated by tunable localized phonons, providing new insights into possible microscopic pairing pathways in quantum paraelectric systems.