Electromagnetic fields can be confined in the presence of metal nanostructures. Recently, subnanometer scale confinement has been demonstrated to occur at atomic protrusions on plasmonic nanostructures. Such an extreme field may dominate atomic-scale light-matter interactions in "picocavities". However, it remains to be elucidated how atomic-level structures and electron transport affect plasmonic properties of a picocavity. Here, using low-temperature optical scanning tunneling microscopy (STM), we investigate inelastic light scattering (ILS) in the vicinity of a single-atom quantum point contact (QPC). A vibration mode localized at the single Ag adatom on the Ag(111) surface is resolved in the ILS spectrum, resulting from tip-enhanced Raman scattering (TERS) by the atomically confined plasmonic field in the STM junction. Furthermore, we trace how TERS from the single adatom evolves as a function of the gap distance. The exceptional stability of the low-temperature STM allows to examine distinctly different electron transport regimes of the picocavity, namely, in the tunneling and QPC regimes. This measurement shows that the vibration mode localized at the adatom and its TERS intensity exhibits a sharp change upon the QPC formation, indicating that the atomic-level structure has a crucial impact on the plasmonic properties. To gain microscopic insights into picocavity optomechanics, we scrutinize the structure and plasmonic field in the STM junction using time-dependent density functional theory. The simulations reveal that atomic-scale structural relaxation at the single-atom QPC results in a discrete change of the plasmonic field strength, volume, and distribution as well as the vibration mode localized at the single atom. These findings give a qualitative explanation for the experimental observations. Furthermore, we demonstrate that strong ILS is a characteristic feature of QPC by continuously forming, breaking, and reforming the atomic contact and how the plasmonic resonance evolves throughout the nontunneling, tunneling, and QPC regimes.