Have you debugged the C# DLL01 program and checked the parameters being passed in both when being called from a console app and being called from a web site?
You state that the web site is an unmanaged COBOL program so I am assuming this is compiled as a CGI .EXE program is that correct? Or is it a multi-threaded ISAPI program? Is there a reason you are using such outdated web technology when VC supports ASP.NET, etc?
I am not sure what you mean by the .dll accessing its configuration file as normally config files such as app.config an web.config are only processed by a main program and not by a .dll. Where is the config file located and how is it accessed.
I don't believe it is the interface between COBOL and C# that is different but more likely how the application is run such as current working directory etc. This could have something to do with the fact that the current working directly that is in play when run as a web app may not be the same as when running as a console app.