I have never seen a way to choose what NIC you want to use per application.
When you say virtual network adaptor, are you talking about a vNIC created through MS Virtual Server or VMware?
What I am thinking is why not use the free VMware player, VMware Server, or MS Virtual PC to run a virtual system, then run your Mozilla inside the virtual guest OS. By doing that, yes, you could easily connect through the vNIC with Mozilla and through the sis900 using IE.
I hope that helps!