We present an accurate computational study of the electronic structure and lattice dynamics of solid molecular hydrogen at high pressure. The band-gap energies of the C2/c, Pc, and P63/m structures at pressures of 250, 300, and 350 GPa are calculated using the diffusion quantum Monte Carlo (DMC) method. The atomic configurations are obtained from ab initio path-integral molecular dynamics (PIMD) simulations at 300 K and 300 GPa to investigate the impact of zero-point energy and temperature-induced motion of the protons including anharmonic effects. We find that finite temperature and nuclear quantum effects reduce the band-gaps substantially, leading to metallization of the C2/c and Pc phases via band overlap; the effect on the band-gap of the P63/m structure is less pronounced. Our combined DMC-PIMD simulations predict that there are no excitonic or quasiparticle energy gaps for the C2/c and Pc phases at 300 GPa and 300 K. Our results also indicate a strong correlation between the band-gap energy and vibron modes. This strong coupling induces a band-gap reduction of more than 2.46 eV in high-pressure solid molecular hydrogen. Comparing our DMC-PIMD with experimental results available, we conclude that none of the structures proposed is a good candidate for phases III and IV of solid hydrogen.