To submit to a document library the user needs to have permissions on the SharePoint library and they have to have at least contribute.
Managing access to documents in SharePoint can be hard sometimes, granular permissions are not always easy.
One thing you could do is to setup a library that is for submit only and that a workflow in SharePoint automatically upon creation of that item the workflow moves it to another library where the user has no access and then the workflow deletes the item from SharePoint.
We also sell a tool called DBXL that has SharePoint mapping built into it. in your situation you would install DBXL on your SharePoint (or another web server) and the DBXL Web Service would be the one with the permissions on the document library. The user would submit to DBXL and DBXL would submit to SharePoint and the user does not need any permissions on the SharePoint library.
Information about DBXL is located here: http://www.qdabra.com/en/products/DBXL.aspx
Ted