We demonstrate nonlinear emission from molecular layers strongly coupled to extended light fields in arrays of plasmonic nanoparticles in the presence of structural imperfections. Hybrid light-matter states, known as plasmon-exciton polaritons (PEPs), are formed by the strong coupling of Frenkel excitons in molecules to surface lattice resonances. These resonances result from the radiative coupling of localized surface plasmon polaritons in silver nanoparticles enhanced by diffraction on the array. By designing arrays with different lattice constants, we show that the nonlinear emission frequency is solely determined by the relaxation of exciton polaritons through vibrational quanta in the molecules. We also observe long-range spatial coherence in the samples, which supports the explanation in terms of a nonlinear collective emission of strongly coupled PEPs. In contrast to recent observations of exciton-polariton lasing and condensation in organic systems, photonic modes play a minor role at the emission frequency in our system, and this emission has an undefined momentum because of the structural imperfections. This remarkable result reveals the rich and distinct physics of strongly coupled organic molecules to photonic cavities.