Arrays of subwavelength plasmonic nanoparticles exhibiting narrowband lattice resonances are referred to as open cavities because of their ability to strongly couple with electronic excitations in molecular chromophores. However, realization of these ideas in the mid-infrared spectral region has been limited. We demonstrated a dramatic reduction in the bandwidth of lattice resonances in large-area arrays of half-wavelength mid-infrared antennas, reaching resonance quality factors above 200. By tuning the wavelength of the antenna-lattice resonances (ALR) to match the transition frequency of the molecular vibrational modes, we achieved a strong coupling between the ALR and the carbonyl stretching excitation in a thin film of (poly)methyl methacrylate (PMMA) polymer deposited on the array. Splitting of the polaritonic transitions, reduction of their bandwidth below that of the bare molecular transition, and characteristic dispersion confirmed the strong coupling regime. Our results pave the way for exciting research on the many-body correlated dynamics of vibrational polaritons.