  We employ the functional renormalization group to study the effects of
phonon-assisted tunneling on the nonequilibrium steady-state transport through
a single level molecular quantum dot coupled to electronic leads. Within the
framework of the spinless Anderson-Holstein model, we focus on small to
intermediate electron-phonon couplings, and we explore the evolution from the
adiabatic to the antiadiabatic limit and also from the low-temperature
non-perturbative regime to the high temperature perturbative one. We identify
the phononic signatures in the bias-voltage dependence of the electrical
current and the differential conductance. Considering a temperature gradient
between the electronic leads, we further investigate the interplay between the
transport of charge and heat. Within the linear response regime, we compare the
temperature dependence of various thermoelectric coefficients to our earlier
results obtained within the numerical renormalization group [Phys.~Rev.~B {\bf
96}, 195156 (2017)]. Beyond the linear response regime, in the context of
thermoelectric generators, we discuss the influence of molecular vibrations on
the output power and the efficiency. We find that the molecular energy
dissipation, which is inevitable in the presence of phonons, is significantly
suppressed in the antiadiabatic limit resulting in the enhancement of the
thermoelectric efficiency.
