jQuery/js are only able to send HTTP demands. You will need to make use of a whois service accessible via http. i.e. a script running in your server, or perhaps an API supplied by another person.
You cannot dependably get it done around the client side only, but when JS is the only option, you should utilize a 3rd party through either an API (for example http://www.whoisxmlapi.com/domain-availability.php) or low volumes of searches, you can easily parse it from another site (e.g. http://whois.domaintools.com/whatever.com)
You will find more integrated and far cleaner server side solutions if that is a choice - phpwhois for instance is thorough and free.