The Green's-function techniques, especially the one developed in the preceding paper [Takada, Phys. Rev. B 52, 12 708 (1995)], are employed to calculate the electron-phonon vertex part as well as the electronic self-energy exactly on both real- and imaginary-frequency axes in the electron-phonon Holstein model with the on-site Coulomb repulsion in the limit in which the intramolecular phonon energy ${\mathrm{\ensuremath{\omega}}}_{0}$ is much larger than the electronic bandwidth. The rigorous vertex part is found to diverge at the frequencies at which an electron is locked by such local phonons with an infinitely strong effective coupling. Characteristic frequencies of this divergence, which are not equal to multiples of ${\mathrm{\ensuremath{\omega}}}_{0}$, are calculated as a function of the electron-phonon bare coupling constant. Our results for the self-energy are checked successfully with the exact ones obtained by the Lang-Firsov canonical transformation.