Strong plasmon–exciton coupling between tightly-bound excitons in organic molecular semiconductors and surface plasmons in metal nanostructures has been studied extensively for a number of technical applications, including low-threshold lasing and room-temperature Bose-Einstein condensates. Typically, excitons with narrow resonances, such as J-aggregates, are employed to achieve strong plasmon–exciton coupling. However, J-aggregates have limited applications for optoelectronic devices compared with organic conjugated polymers. Here, using numerical and analytical calculations, we demonstrate that strong plasmon–exciton coupling can be achieved for Ag-conjugated polymer core-shell nanostructures, despite the broad spectral linewidth of conjugated polymers. We show that strong plasmon–exciton coupling can be achieved through the use of thick shells, large oscillator strengths, and multiple vibronic resonances characteristic of typical conjugated polymers, and that Rabi splitting energies of over 1000 meV can be obtained using realistic material dispersive relative permittivity parameters. The results presented herein give insight into the mechanisms of plasmon–exciton coupling when broadband excitonic materials featuring strong vibrational–electronic coupling are employed and are relevant to organic optoelectronic devices and hybrid metal–organic photonic nanostructures.