I’m sure you’ve read through these responses and seen already, but what you want to do is impossible through only client-side code. These server-side solutions work, because alot of these server environments have the different servers and whatnot set up to handle sending off an email.
It’s not as simple as doing an AJAX request.
You may be able to find a third party service that helps you handle transactional email, like Mandrill.
These work, because they do all the work that the server-side needs. All you need to do is access their API endpoint through AJAX the way you would anything else, and bam. Mandrill will shoot off your email for you.
Now, Mandrill has a free tier, and that may be all you need. But think about how much this will be used, as it may just be more worth it to use something like Mandrill, as opposed to rolling your own sever side solution.