Ajax was there way before the advent of ASP.net so the answer is "yes" one can use Ajax without ASP.net.
You can write HTML utilizing Ajax from scratch only using JavaScript, but there are some JavaScript frameworks out there that make the task so much easier:
These are only two frameworks, but there are others as well, which you can research using google. I even had a solution a way back that listed different JavaScript frameworks. If I find it again I'll post a link here. (Hang on! :) )
[Edit]
I'm sorry to say the question with my previous solution was deleted. Nevertheless a short google session turned up this little gem:
The most complete Ajax framework and JavaScript libraries list.html[
^]
There are 124+ frameworks & libraries listed there.
[/Edit]
Regards,
—MRB