The wave functions of a disordered two-dimensional electron gas at the quantum-critical Anderson transition are predicted to exhibit multifractal scaling in their real space amplitude. We experimentally investigate the appearance of these characteristics in the spatially resolved local density of states of the two-dimensional mixed surface alloy BixPb(1-x)/Ag(111), by combining high-resolution scanning tunneling microscopy with spin- and angle-resolved inverse-photoemission experiments. Our detailed knowledge of the surface alloy's electronic band structure, the exact lattice structure, and the atomically resolved local density of states enables us to construct a realistic Anderson tight binding model, and to directly compare the measured local density of states characteristics with those from our model calculations. The statistical analyses of these two-dimensional local density of states maps reveal their log-normal distributions and multifractal scaling characteristics of the underlying wave functions with a finite anomalous scaling exponent. Finally, our experimental results confirm theoretical predictions of an exact scaling symmetry for Anderson quantum phase transitions in the Wigner-Dyson classes.