Yes, a WebClient can work well for this. The WebBrowser control will work as well depending on your requirements. If you are going to load the document into a HtmlDocument (the IE HTML DOM) then it might be easier to use the web browser control.
The HtmlDocument object that is now built into .NET can be used to parse the HTML. It is designed to be used with the WebBrowser control but you can use the implementation from the mshtml dll as well. I hav enot used the HtmlAgilityPack, but I hear that it can do a similar job.
The HTML DOM objects will typically handle, and fix up, most ugly HTML That you throw at them. As well as allowing a nicer way to parse the html, document.GetElementsByTag to get a collection of tag objects for example.
As for handling the changing requirements of the site, it sounds like a good candidate for the strategy pattern. You could load the strategies for each site using reflection or something of that sort.
I have worked on a system that uses XML to define a generic set of parameters for extracting text from HTML pages. Basically it would define start and end elements to begin and end extraction. I have found this technique to work well enough for a small sample, but it gets rather cumbersome and difficult to customize as the collection of sites gets larger and larger. Keeping the XML up to date and trying to keep a generic set of XML and code the handle any type of site is difficult. But if the type and number of sites is small then this might work.
One last thing to mention is that you might want to add a cleaning step to your approach. A flexible way to clean up HTML as it comes into the process was invaluable on the code I have worked on in the past. Perhaps implementing a type of pipeline would be a good approach if you think the domain is complex enough to warrant it. But even just a method that runs some regexes over the HTML before you parse it would be valuable. Getting rid of images, replacing particular mis-used tags with nicer HTML , etc. The amount of really dodgy HTML that is out there continues to amaze me...