I have a Google Apps account with two others approved "Send mail as" accounts. In the Gmail settings (web interface) I can set one signature for each of the three email addresses.
Is it possible to do the same through the Googe Apps Email Settings API so that I can automate this for the entire organization?
The API documentation (https://developers.google.com/google-apps/email-settings/#manage_signature_settings) does only mention how to define username, but gives no options for specifying the actual email address for that username to set the signature for.
It's not currently possible, no. The Email Settings API only supports setting the signature for the primary email address.