We propose a cavity QED approach to describe light-matter interaction of an infrared cavity field with an anharmonic vibration of a single nonpolar molecule. Starting from a generic Morse oscillator potential with quantized nuclear motion, we derive a multilevel quantum Rabi model to study vibrational polaritons beyond the rotating-wave approximation. We analyze the spectrum of vibrational polaritons in detail and compare it with available experiments. For high excitation energies, the system exhibits a dense manifold of polariton level crossings and avoided crossings as the light-matter coupling strength and cavity frequency are tuned. We also analyze polariton eigenstates in nuclear coordinate space. We show that the bond length of a vibrational polariton at a given energy is never greater than the bond length of a Morse oscillator with the same energy. This type of polariton bond strengthening occurs at the expense of the creation of virtual infrared cavity photons and may have implications in chemical reactivity of polariton states.