Normally you'd have a PHP/python/perl/ruby/node/whatever script listening on the server for an HTTP POST (or AJAX request, or RPC, or websocket, or whatever) that would send the mail for you. But this won't work here because S3.
The trick is to find someone who'll run a server for you that will accept messages over an HTTP interface and pass them along to a designated email address. And apparently Squaresend will do this, which is honestly pretty nice if you just want a contact form on your S3 hosted website (which is not that strange a need!).
Of course it is a hosted solution just like squaresend, just another alternative to consider.