Above I suggest that’s only possible to process the data in a form by using a third party service. Of course this isn’t completely true because it’s still possible to add a PHP script to your website (for example in a subdomain of your website). If this is what you want to do then PHPMailer is probably your best option.
You can find more information about PHPMailer here:
https://github.com/PHPMailer/PHPMailer/blob/master/README.md
