There are plenty of ways to do this; the most common revolve around AJAX. You can initiate a callback from the client via Javascript to a method on the server, which can update controls in an UpdatePanel, for example.
You can also make client side calls to server side Page Methods. Effectively, this is a static method on your webform that you can call from the client via javascript/jquery and AJAX.
EDIT.
It turns out that you want to scrape another site. The easiest way for you to do this is have a server side page method on your website that does this - it requests the page from the client site, extracts the info you want, and then returns that to your client. Your client can of course call this as a page method.
See https://web.archive.org/web/20210513000146/http://www.4guysfromrolla.com/webtech/070601-1.shtml for a tutorial, and I do suggest using the HTML Agility Pack as that article mentions.
Further EDIT
You want to further manipulate the page on the remote site; if you can't or don't want to speak to the developers of that site to work out a way of doing it programmatically, then you'll have to cheat. Get Firebug and Tamper Data. Use Firebug and Tamper Data to see how clicking the button on the remote site makes a request and posts it to the server - you want to emulate doing the same. If you know what data is being posted then you can, from your server, make exactly the same post.
You often have this kind of problem when trying to scrape AJAX websites.