We present direct experimental evidence of Anderson localization induced by the intrinsic alloy compositional disorder of InGaN/GaN quantum wells. Our approach relies on the measurement of the luminescence spectrum under local injection of electrons from a scanning tunneling microscope tip into a near-surface single quantum well. Fluctuations in the emission line shape are observed on a few-nanometer scale. Narrow emission peaks characteristic of single localized states are resolved. Calculations in the framework of the localization landscape theory provide the effective confining potential map stemming from composition fluctuations. This theory explains well the observed nanometer scale carrier localization and the energies of these Anderson-type localized states. The energy spreading of the emission from localized states is consistent with the usually observed very broad photo- or electroluminescence spectra of InGaN/GaN quantum well structures.