The quantum dynamics of atoms in amorphous silicon has been addressed by using path-integral Monte Carlo simulations. Structural results (radial distribution functions) found from these simulations agree well with experimental data. We study the quantum delocalization of the silicon atoms around their equilibrium positions. This delocalization is larger for coordination defects (fivefold-coordinated Si atoms). Correlations in the atomic displacements are analysed as a function of the interatomic distance and compared with those derived from classical Monte Carlo simulations. At high temperatures, the classical limit is recovered. Our results are also compared with those derived from similar quantum simulations for crystalline silicon. Structural disorder favours a larger vibrational amplitude for the atoms in amorphous silicon.