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A WCF service is a possible solution:
http://msdn.microsoft.com/en-us/library/ms734712.aspx
Windows has scheduled tasks which is good at... scheduling tasks.
Do you (or plan to) have a big test of GUI tests? There are entire tools dedicated to GUI automation testing. I'd recommend looking into one of those if this is going to be a big part of your overall test strategy.
If this is a one-time thing, you could schedule the running of a simple C# application that hosts a Web Browser control, that points to your web site. In this sense, the Web Browser control acts as your browser. You can send JavaScript commands to it etc.
Web Browser documentation: http://msdn.microsoft.com/en-us/library/system.windows.forms.webbrowser.aspx
Another alternative is to schedule an AutoHotKey script to simulate the key pressing and mouse clicks. This works well. I used to use it to rack up Farmville points (no joke).
http://www.autohotkey.com/
A Windows service seems better in the sense of "scheduled" and "no intervention." A WCF service is still, by default, listening and waiting for interaction (hosting the service). Or just schedule a simple app or script to do what you need.
Can you tell us a little more about this process? My initial recommendation would be to transition that code from a web application into a command line utility and then use windows task scheduler. If for some reason there are heavy dependencies within that web app that make that impossible I would consider taking those button click events and turning them into web services that you could then call programmatically from a command line application.