A novel type of exciton–phonon bound state—interlayer polaron—in a double-layer 2D 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.