You can look at your web application as one with schizophrenia...
It has two parts - one that runs at the server and the other that runs on the client...For the argument you can say that the server side is no different from a win application...And I mean that you can access any external code (your WIN32 based dll) the exact same way you do in your win application...
And now that's what brings you your problem with support Linux based web server...You can run ASP.NET applications in
Apache[
^] server using the proper mod, but the WIN32 based code may have some problems to run...You will have to test it with
wine (or alernatives)[
^] to see if it is possible to run that specific code at all...