You could install PHP on the laptop with that serial class, and have the website they visit use curl or sockets to the server on the laptop to trigger the serial class.
You'd go Client Laptop -> Webserver -> Client Laptop
Client laptop would send the request to the webserver, the webserver would then send it's OWN request to the CLIENT laptop's server using something like curl or a socket connection. The client laptop would have code snippet that would catch the incoming request, then run the serial class code on its own. PHP may not be the best language for this, but if you figure it out, please let me know. 🙂