Neutron diffraction data is the best method, but not always available. However, it is possible to get oxygen occupancy info in many cases from X-Ray powder diffraction data. Check out this reference:

Zhu, Y. T., et al. (1998). "Variation of oxygen content and crystal chemistry of YBa4Cu3O8.5+delta." Physica C 298(1-2): 29-36.

In some materials, e.g. Perovkites there is a relation between oxygen content and cell parameters. Some times phase transitions are observed for different oxygen contents (as is the case of LaMnO3-d) and other times there is a slight change in cell parameters due to different average bond distances among the cations when O vacancies are introduced). This is not considering a Rietveld refinement where yo can determine the number and ordering of vacancies (if that were the case) with both neutron or x-ray diffraction (the first preferred over the second an high resolutiin synchrotron x-rays preferred over conventional diffractometer data).

With conventional x-ray powder diffraction data it is almost impossible to detemine oxygen content accurately in the presence of transition metals or heavier atoms since small errors due to the simplicity of the spherical atom model are probably of the same order of your electronic density differenc due to vacancies, beside other instumental contribution to electron density errors.

The intensity of scattered x-ray is a function of atomic number (Z). Electrons are source of x-ray scattering, thus as Z increases, the number of electrons within an atom increase and intensity of scattered x-ray by an atom increases. Thus in presence of High Z-element like (Mn, Fe, Ti etc) the contribution to overall scattering intensity by low Z elements (e.g Oxygen) will be very small and insufficient to calculate concentration of low-Z element (like oxygen) in a system containing high Z elements. Thus x-ray diffraction is not a correct tool for it.

But neutron scattering cross section for oxygen is large and by Rietveld refinement of neutron diffraction you can calculate the oxygen atomic and mass percentage in your materials accurately and then knowing the chemical formula of your materials you can calculate the concentration of oxygen vacancies.