I am sorry if my reply is not positive enough, but as far as I know, it is not possible to automate WPF gui with webdriver. For web part, assuming a browser app displayed to user in a form of markup ja javascript, you definitely can use webdriver.
I had an automation project back in 2011-2012 where I had to automate a .NET desktop application (gui in WPF) and it was not nice at all. If you have money, you can try using the CodedUI from microsoft but .... it was not a state of art, at least not back then.. What you have is testing frameworks built upon accessibility frameworks. Anyway, (assuming you have access to the source,) I managed to get somewhat tolerable results using System.Windows.Automation namespace (
http://msdn.microsoft.com/en-us/library/system.windows.automation.aspx), although it did require implementing the AutomationPeer on each and every custom control. Anyway, depending on how much you need, you can easily just send the UIElement.RaiseEvent() with MouseButtonEventArgs (like mouse down routed event, or keyboard).
Or you mean those "browser based WPF apps (xbap)?" - i know nothing of them, sry.
Good luck
erki.