In quantum electrodynamics, matter can be hybridized to confined optical fields by a process known as light-matter strong coupling. This gives rise to new hybrid light-matter states and energy levels in the coupled material, leading to modified physical and chemical properties. Here, we report for the first time the strong coupling of vibrational modes of proteins with the vacuum field of a Fabry-Perot mid-infrared cavity. For two model systems, poly(l-glutamic acid) and bovine serum albumin, strong coupling is confirmed by the anticrossing in the dispersion curve, the square root dependence on the concentration, and a vacuum Rabi splitting that is larger than the cavity and vibration line widths. These results demonstrate that strong coupling can be applied to the study of proteins with many possible applications including the elucidation of the role of vibrational dynamics in enzyme catalysis and in H/D exchange experiments.