Surface-enhanced vibrational spectroscopy strongly increases the cross section of Raman scattering and infrared absorption, overcoming the limited sensitivity and resolution of these two powerful analytic tools. While surface-enhanced setups with maximum enhancement have been studied widely in recent years, substrates with reproducible, uniform enhancement have received less attention although they are required in many applications. Here, we show that plasmonic supercrystals are an excellent platform for enhanced spectroscopy because they possess a high density of hotspots in the electric field. We describe the near field inside the supercrystal within the framework of plasmon polaritons that form due to strong light-matter interaction. From the polariton resonances we predict resonances in the far-field enhancement for Raman scattering and infrared absorption. We verify our predictions by measuring the vibrations of polystyrene molecules embedded in supercrystals of gold nanoparticles. The intensity of surface-enhanced Raman scattering is uniform within 10% across the crystal with a peak integrated enhancement of up to 300 and a peak hotspot enhancement of 105. The supercrystal polaritons induce pairs of incoming and outgoing resonances in the enhanced cross section as we demonstrate experimentally by measuring surface-enhanced Raman scattering with multiple laser wavelengths across the polariton resonance. The infrared absorption of polystyrene is likewise enhanced inside the supercrystals with a maximum enhancement of 400%. We show with a coupled oscillator model that the increase originates from the combined effects of hotspot formation and the excitation of standing polariton waves. Our work clearly relates the structural and optical properties of plasmonic supercrystals and shows that such crystals are excellent hosts and substrates for the uniform and predictable enhancement of vibrational spectra.