We describe the two dimensional angular correlation of positron annihilation radiation (2D-ACAR) method and show that it can be advantageously used to study the electronic structure of defects, in addition to standard positron lifetime and Doppler broadening measurements. Using annihilation fractions determined by lifetime measurements, we separate 2D-ACAR distributions for negatively charged and neutral arsenic vacancies in n-type GaAs. In electron-irradiated semi-insulating GaAs, we present 2D-ACAR results for the negatively charged gallium vacancy and for positron Rydberg states induced by gallium antisites. The 2D-ACAR for delocalized positrons, needed in the separation process, has been obtained from measurements on as-grown semi-insulating GaAs. Our results outline the capabilities of the 2D-ACAR technique when it is applied to the study of defects: the electronic structure in the vacancies are found to be well described by recent molecular dynamics calculations. The method also provides information about atomic relaxations around the vacancy sites, they are in agreement with the calculated relaxations. We conclude that the 2D-ACAR method is a promising tool to investigate defects in semiconductors.