Adobe Sign API questions

Out of curiosity, I'm wondering if the API is/has a method of calling for stored finished forms on the Adobe Cloud. So for example, if I have my own hosted website and want to have a set of fillable forms for our users to complete and submit, can I have the embedded forms filled out, submitted, then "grab" the info they submitted to be stored on our database or grab the files/data from adobe without having to do so manually? When I spoke to support, they made it sound like I would literally be forced to manually save every single submitted form to my hard drive, but that seems counter-intuitive and the very fact an API exists makes me think otherwise.

Hopefully my question makes sense, thank you in advance.