The unique response of amorphous ionic oxides to changes in oxygen stoichiometry is investigated using computationally intensive ab initio molecular dynamics simulations, comprehensive structural analysis, and hybrid density-functional calculations for the oxygen defect formation energy and electronic properties of amorphous In2O3-x with x=0-0.185. In marked contrast to nonstoichiometric crystalline nanocomposites with clusters of metallic inclusions inside an insulating matrix, the lack of oxygen in amorphous indium oxide is distributed between a large fraction of undercoordinated In atoms, leading to an extended shallow state for x0.185. The calculated carrier concentration increases from 3.3x1020cm-3 at x=0.037 to 6.6x1020cm-3 at x=0.074 and decreases only slightly at lower oxygen content. At the same time, the density of deep defects located between 1 and 2.5 eV below the Fermi level increases from 0.4x1021cm-3 at x=0.074 to 2.2x1021cm-3 at x=0.185. The wide range of localized gap states associated with various spatial distributions and individual structural characteristics of undercoordinated In is passivated by hydrogen that helps enhance electron velocity from 7.6x104 to 9.7x104 m/s and restore optical transparency within the visible range; H doping is also expected to improve the material's stability under thermal and bias stress.