We present a procedure for addressing extrinsic defects in amorphous oxides, in which the most stable defect configurations are identified through ab initio molecular dynamics in various charge states and studied through hybrid functional calculations. The protocol is further complemented with an electron counting scheme based on maximally localized Wannier functions, which allows one to identify the nominal charge state and the composition of the defect core unit. Here, we apply this approach to the study of hydrogen, carbon and nitrogen impurities in amorphous alumina (alpha-Al2O3), a highly relevant material for technological applications. Hydrogen is found to be amphoteric with a thermodynamic +1/-1 charge transition level lying at similar to 4.6 eV above the valence band maximum, in qualitative agreement with results obtained with crystalline models, but at a substantially different energy level. Hydroxyl groups are further shown to lead to the same defect states as observed for hydrogen. Application of our procedure to carbon and nitrogen impurities leads to structural configurations that are found to depend on the total charge set in the simulation cell. Through the adopted electron counting rule, we assess that carbon and nitrogen impurities are only found in neutral and in singly positive charge states, respectively. This indicates that neither carbon nor nitrogen give charge transition levels in the band gap, in strong contrast with results achieved for crystalline models. In addition, the defect core units are shown to incorporate a varying number of oxygen atoms, by which their formation energy depends on the oxygen chemical potential mu O. In oxygen-poor conditions, both the carbon and the nitrogen impurities favor bonding to Al atoms, while they tend to form single or double bonds with oxygen atoms as mu(0) increases. Based on the band alignment of alpha-Al2O3 with three technologically relevant semiconductors (GaAs, GaN and alpha-Fe2O3), we discuss the possible role of point defects in degrading the performance of electronic devices and in favoring hole transport across the oxide in water-splitting setups.