Question: Is it possible to upload files to a web browser application (such as Dropbox) directly from a document management system (DMS) such as iManage (aka Worksite, Filesite, HP Autonomy)? 

If not, what are the best ways of getting around this and enabling simple upload of files from the DMS to web applications?

Answer: 
If you are trying to share documents via a dropbox like solution, you might want t consider LinkSite. It is more manageable, which is quite a positive aspect since you will probably working in a legal environment. 

More info can be found here: <http://www.irisecm.com/hp-linksite-the-secure-and-enterprise-alternative-to-dropbox/>
Best regards,
Daniel
By the way, LinkSite is rebranded and now called iManage Share. 
If you are looking for a seamlessly integrated third party application you might want to consider Litéra Sync.