Vibrational ultrastrong coupling, where the light-matter coupling strength is comparable to the vibrational frequency of molecules, presents new opportunities to probe the interactions between molecules and zero-point fluctuations, harness cavity-modified chemical reactions and develop novel devices in the mid-infrared spectral range. Here we use epsilon-near-zero nanocavities filled with a model polar medium (SiO2) to demonstrate ultrastrong coupling between phonons and gap plasmons. We present classical and quantum-mechanical models to quantitatively describe the observed plasmon-phonon ultrastrong coupling phenomena and demonstrate a modal splitting of up to 50% of the resonant frequency (normalized coupling strength eta > 0.25). Our wafer-scale nanocavity platform will enable a broad range of vibrational transitions to be harnessed for ultrastrong coupling applications.