Well, this is a bit of a bigger piece of architecture than I was expecting so I am not sure what is available to you to change/implement, but here is what I would do...
Put your vbscript variable into a form input variable. When the user submits the page that variable is transmitted to the server. This is a fairly standard usage for client-side VBScript so that should be straight-forward.
On the server side, you need to pick (or let me know what) server side technology you are using. If you are wide open, I would recommend using ASP.NET, but I am a .NET advocate.
However you decide to process the server request, you are going to take the parameter sent from the client's browser and use the LabVIEW ActiveX Server interface (see User's Manual for more info) to load up and run the VI.
If this sounds like it does the job, let me know and I can answer more specific questions for the various stages. If this doesn't work, I'm going to need a bit more information to help.
BTW - There is a small problem running LV from ASP.NET that requires a bit-o-code to get around in the current release, so if we are going that way, let me know and I'll tell you about it.