  We show that in presence of a deformable lattice potential, the nature of the
disorder-driven metal-insulator transition (MIT) is fundamentally changed with
respect to the non-interacting (Anderson) scenario. For strong disorder, even a
modest electron-phonon interaction is found to dramatically renormalize the
random potential, opening a mobility gap at the Fermi energy. This process,
which reflects disorder-enhanced polaron formation, is here given a microscopic
basis by treating the lattice deformations and Anderson localization effects on
the same footing. We identify an intermediate "bad insulator" transport regime
which displays resistivity values exceeding the Mott-Ioffe-Regel limit and with
a negative temperature coefficient, as often observed in strongly disordered
metals. Our calculations reveal that this behavior originates from significant
temperature-induced rearrangements of electronic states due to enhanced
interaction effects close to the disorder-driven MIT.
