We investigate the critical supercurrent in Josephson junctions consisting of topological Anderson insulators (TAIs) via the Matsubara Green's function formalism. Our numerical results show that the disorder-induced edge states display distinct differences in dominating normal and supercurrent transport in the TAI phase. Unlike the hallmark of the TAI phase which exhibits a quantized conductance plateau, the critical supercurrent over the disorder strength exhibits a peak structure where the maximum value is reached at the weak-disorder boundary of the TAI phase. Although the magnitude of the averaged critical supercurrent is suppressed with increasing disorder strength, periodic oscillations of the supercurrent on an external magnetic flux survive in the TAI phase. These findings indicate that the supercurrent quantum interference effect can be an effective probe in detecting the emergence of disorder-induced edge state in TAIs.