A single emitter can couple with electromagnetic modes of dielectric cavities or metallic particles. In a similar manner, it can couple with a phononic mode supported by a nearby infrared antenna. We consider an emitter with a sufficiently large dipole moment coupled to a SiC bowtie structure supporting strongly localized phononic modes. We show that vacuum Rabi oscillations and large spectral anticrossing are possible, indicating that the emitter-phononic system is in the strong coupling regime. Pure dephasing degrades the response remarkably little. As expected for a quantum but not for a classical formalism, the frequency of the vacuum Rabi oscillations depends on the initial state. We also discuss the possibility of exciting hybrid modes with contributions from the emitter and from more than one of the phononic modes supported by the antenna. Phononic structures appear attractive to study such complex hybridization, as they can support several strongly confined modes with quality factors larger than one hundred in a relatively small spectral window.