Geocodio can return school district information for US addresses and coordinate pairs. You can upload a spreadsheet of addresses or coordinates, or use our API.
Just select "School districts" in Step 3 of the upload process.
School district data returned:
In the case of a unified district, this information will be returned once. For areas with separate elementary and secondary districts, the information will be returned for both districts.
You can download an example spreadsheet of addresses with school districts districts added here.