If you're not adverse to cloud file storage, FormKiQ Core (I'm a co-founder) is an open source document management system that runs on AWS and is designed to allow custom integrations.
This looks very nice. I wish the price point for Formkiq Pro were a little more palatable for a startup.
I'm one of the co-founders of FormKiQ. We had originally expected startups to work with Core (the free version), but based on feedback we're looking at some changes, including a startup program. Let me know if you'd like to discuss how this could work. Thanks!