When do we use: File with and file at

For instance, I want an agency or company to receive my documents. Do you say, file with or file at?

Neither, I think. Please supply a complete sentence.

For example: This document should be filed with the State Department. Is this correct? On the other hand, can you say: This document should be filed at the State Department? Is this also correct?

OK-- either will work. ‘With’ seems better to me.

…filed with an organization, …sent to an organization