JavaScript needs access to cookies if AJAX is used on a site with access restrictions based on cookies. Will HttpOnly cookies work on an AJAX site?
Edit: M
Therefore I am assuming JavaScript needs access to your cookies.
All HTTP requests from your browser transmit your cookie information for the site in question. JavaScript can both set and read cookies. Cookies are not by definition required for Ajax applications, but they are required for most web applications to maintain user state.
The formal answer to your question as phrased - "Does JavaScript need access to cookies if AJAX is used?" - is therefore "no". Think of enhanced search fields that use Ajax requests to provide auto-suggest options, for example. There is no need of cookie information in that case.