Exciton-polariton condensation in organic materials, arising from the coupling of Frenkel excitons to the electromagnetic field in cavities, is a phenomenon resulting in low-threshold coherent light emission among other fascinating properties. The exact mechanisms leading to the thermalization of organic exciton-polaritons toward condensation are not yet understood, partly due to the complexity of organic molecules and partly to the canonical microcavities used in condensation studies, which limit broadband studies. Here, we exploit an entirely different cavity design, i.e., an array of plasmonic nanoparticles strongly coupled to organic molecules, to successfully measure the broadband ultrafast dynamics of the strongly coupled system. Sharp features emerge in the transient spectrum originating from the formation of a condensate with a well-defined molecular vibrational composition. These measurements represent the first direct experimental evidence that molecular vibrations drive condensation in organic systems and provide a benchmark for modeling the dynamics of organic-based exciton-polariton condensates.