Amorphous hafnium dioxide (a-HfO$_2$) is widely used in electronic devices, such as ultra-scaled field-effect transistors and resistive memory cells. The density of oxygen vacancy (OV) defects in a-HfO 2 strongly influences the conductivity of the amorphous material. Ultimately, OV defects are responsible for the formation and rupture of conductive filament paths which are exploited in novel resistive switching devices. In this work, we studied neutral OV in a-HfO$_2$ using ab initio methods. We investigated the formation energy of OV, the binding energy of di-OVs, the OV migration, unperturbed and in the close presence of a hydrogen atom, as well as the migration of hydrogen atom towards OV. A shallow and short-range OV migration barrier (0.6 eV) exists in a-HfO$_2$ in contrast to the barrier (2.4 eV) in crystalline HfO 2 . Nearby hydrogen has a limited impact on the OV migration; however, hydrogen can diffuse easily by hopping among OVs.