No, your server application is not Firefox browser.
All browsers (IE, Firefox, Chrome, Opera, ...) run oin the client, and communicate with a server based application running in a webserver (another program that understands how to manaage websites).
Even if you include a web browser in your WinForms application - and that's easy to do, just drop a WebBrowser control on it) it still runs at the client, never the server.
Servers are unattended, so browsers do not make any sense there.
If you want to watch a film on your client, you have to send the video data in some form - which in your scenario means "sending the temporary files" to the client.
The alternative is
Streaming[
^]: where the video is "guided" by the server from the provider to the client but the data doesn't go via the server in the first place.