As was noted above, the .NET browser control cannot be used in this case, unless what you want is to open the browser window on your SERVER, since that's where the code is running.
On the client, it's not even .NET code you are interacting with - it's a plain-old Win32 binary. You cannot 'script' the client from the server (and in fact, the UO client can't really be 'scripted' at all, beyond the simple macros built in.) The packet referenced above to get the client to open the user's own web browser is the only way to really do what you want here, without having to have external software involved.
So you mentioned the solution yourself, if you absolutely need to do this; Force your players to download a separate program that you write in .NET, and have that program somehow open or view the page you want to display.
But it would probably just be much easier just to use the uo client's built in browser opening functionality, and just make the page you are pointing to compatible with all the major browsers, so you don't need to box your players into needing a particular browser.
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Also, just for the record; the 'built in' .NET web browser control is actually just whatever current version of Internet Explorer the user has installed, without the normal chrome of the browser. The control is used in a Windows Forms program to implement a browser within the program. It is not a standalone web browser on it's own.