Personally I don't see why this isn't possible. I'm not sure what the above response means by "network based".
You would need to convert the example into an ActiveX server or wrap it in one. Or you might even be able to rewrite the code directly as ASP script.
The .NET framework would probably make this fairly trivial.
EDIT:
I guess I may have been confused about what you are trying to do. For some reason I thought there was a VB sample that came with the TRILOGi software or something.
If you are looking to interact with the PLC using a web page then you will need quite a bit more than the VB serial port sample. You will have to implement the MODBUS protocol that the PLC supports. You also might be able to purchase a third-party OPC server control that knows how to interface with MODBUS devices and can be called from an ASP page. Still not simple, though.