We investigate a symmetry protected Z2 topological electron glass -a glassy equivalent of the Z2 topological band insulator in crystalline systems-and uncover associated quantum phase transitions in this three dimensional amorphous network of atoms. Through explicit numerical calculations of the Witten effect, we show that the Z2 glass is characterized by an anomalous electromagnetic response-dyons with 1/2 electronic charge. We further study, using a variety of numerical diagnostics including such electromagnetic responses, the phase transitions of the Z2 glass in to a metallic and/or a trivial insulating phase. We find that the phase transitions here are governed by subtle features of mobility edges and "spectral inversion" which are possibly unique to structurally amorphous systems. Our results provide a concrete setting to understand the general underpinnings of such phases -where strong structural disorder interplays with symmetry-protected topological order.