How do I send email with JavaScript without opening the mail client?

You cannot cause the user's browser to send email silently. That would be a horrible security problem as any website could use their system as a spam relay and/or harvest their email address.

You need to make an HTTP request to a server side process (written in the language of your choice) which sends the mail from your server.

You need a server-side support to achieve this. Basically your form should be posted (AJAX is fine as well) to the server and that server should connect via SMTP to some mail provider and send that e-mail.

Even if it was possible to send e-mails directly using JavaScript (that is from users computer), the user would still have to connect to some SMTP server (like gmail.com), provide SMTP credentials, etc. This is normally handled on the server-side (in your application), which knows these credentials.