Complex oxide functionality, such as ferroelectricity, magnetism or superconductivity is often achieved in epitaxial thin-film geometries. Oxygen vacancies tend to be the dominant type of defect in these materials but a fundamental understanding of their stability and electronic structure has so far mostly been established in the bulk or strained bulk, neglecting interfaces and surfaces present in a thin-film geometry. We investigate here, via density functional theory calculations, oxygen vacancies in the model system of a SrMnO3 (SMO) thin film grown on a SrTiO3 (STO) (001) substrate. Structural and electronic differences compared to bulk SMO result mainly from undercoordination at the film surface. The changed crystal field leads to a depletion of subsurface valence-band states and transfer of this charge to surface Mn atoms, both of which strongly affect the defect chemistry in the film. The result is a strong preference of oxygen vacancies in the surface region compared to deeper layers. Finally, for metastable oxygen vacancies in the substrate, we predict a spatial separation of the defect from its excess charge, the latter being accommodated in the film but close to the substrate boundary. These results show that surface and interface effects lead to significant differences in stability and electronic structure of oxygen vacancies in thin-film geometries compared to the (strained) bulk.