Multiphonon resonant Raman scattering in self-assembled quantum disks is investigated using a nonadiabatic approach. The optical phonons and the electron-phonon interaction are considered within the multimode dielectric continuum model. The model exploits both electrostatic and mechanical boundary conditions for the relative ionic displacement vector, as well as the phonon spatial dispersion in bulk. The confined phonon modes in a quantum dot are hybrids of bulklike and interface vibrations. It is shown that nonadiabatic effects substantially increase the Raman scattering probabilities and the relative multiphonon integral intensities with respect to the one-phonon intensity. The calculated ratio of the two- and one-phonon integral intensities is close to the experimental value for self-organized InAs GaAs quantum dots.