Well, wouldn't you just have to write a program in whichever language you choose (VB, Perl, Visual C++, etc...) that can be called from within Remedy that accepts certain parameters? (ie, Ticket ID, Status, etc...) Then, have your new program make the necessary changes within your document management system. So, in reality, two separate programs... one that pushes information into Remedy using the Remedy API and another that accepts information (via parameters) from Remedy and pushes it into your document management system using its API...
Make sense?