  We present an extension of the tunneling theory for scanning tunneling
microcopy (STM) to include different types of vibrational-electronic couplings
responsible for inelastic contributions to the tunnel current in the
strong-coupling limit. It allows for a better understanding of more complex
scanning tunneling spectra of molecules on a metallic substrate in separating
elastic and inelastic contributions. The starting point is the exact solution
of the spectral functions for the electronic active local orbitals in the
absence of the STM tip. This includes electron-phonon coupling in the coupled
system comprising the molecule and the substrate to arbitrary order including
the anti-adiabatic strong coupling regime as well as the Kondo effect on a free
electron spin of the molecule. The tunneling current is derived in second order
of the tunneling matrix element which is expanded in powers of the relevant
vibrational displacements. We use the results of an ab-initio calculation for
the single-particle electronic properties as an adapted material-specific input
for a numerical renormalization group approach for accurately determining the
electronic properties of a NTCDA molecule on Ag(111) as a challenging sample
system for our theory. Our analysis shows that the mismatch between the
ab-initio many-body calculation of the tunnel current in the absence of any
electron-phonon coupling to the experiment scanning tunneling spectra can be
resolved by including two mechanisms: (i) a strong unconventional Holstein term
on the local substrate orbital leads to reduction of the Kondo temperature and
(ii) a different electron-vibrational coupling to the tunneling matrix element
is responsible for inelastic steps in the $dI/dV$ curve at finite frequencies.
