AJAX doesn't need a form at all, it is a way of sending a request from the browser to the server which can contain any data you want. For instance, you can track where the user moves their mouse, and send this data to your server every few milliseconds.

If you are using it to send the contents of a form, then technically no, you don't need action and method attributes, because you can read the form fields directly, and send them wherever you like. However, it's usually a good idea to have them anyway, if only as a fallback for if the JavaScript goes wrong, or the user has JavaScript switched off.