We develop a strong-coupling approach to investigate the Anderson-Holstein model with strong repulsion on impurity centers. We derive the relation between electron propagators and correlation functions and prove that for the impurity electrons the latter is identical to the mass operator of the conduction electrons. Strong electron-phonon interaction determines formation of polarons with heavy clouds of phonons surrounding impurity electrons. We demonstrate the existence of a collective excitation mode of these clouds and obtain the dependence of its energy on the hybridization of the impurity with band states. Hybridization is shown to cause softening of the collective mode and its total suppression at sufficiently large values of the hybridization parameter.