A novel type of exciton-phonon bound state -interlayer polaron -in a double-layer twodimensional semiconductor with transition metal dichalcogenides as an example, is predicted. In these systems the interaction of the interlayer exciton with the soft modes of out-of-plane lattice vibrations caused by van der Waals forces and flexural rigidity gives rise to a bound quasiparticle. The energy and effective mass of the formed polaron for weak and strong exciton-phonon coupling regimes are calculated and analyzed. Possible manifestations of these effects in transport-and spectroscopy-related experiments are discussed.