Can you use jQuery with webforms? Yes, that's not a problem - at the end of the day the whole webforms setup generates HTML web pages and hence jQuery can manipulate those.
Can jQuery provide everything the default AJAX controls provide - hmmm, yes in theory because ultimately all those are doing are a) manipulating the DOM and b) making out of band calls to the server. In practice you're going to be giving up some ease of use and you're going to have to do rather more work yourself.
The biggest problem - at least with apps running prior to .NET 4.0 - is that the IDs generated for server controls are, erm, interesting. If you can do everything you need in jQuery without reference to a control ID (if you can use classes or look for control types) you're away and running but if you need to be explicit about an ID you're going to be stressed unless you can use .NET 4.0
I've got code that mixes bits of the two (dynamic data sites that use the jQueryUI calendar as its vastly better than the toolkit one) and would certainly contemplate using more jQuery stuff in forms apps.