It is possible. You don't say if you have PHP, ASP or JSP on your website. They are server side programming languages which can dynamically generate output.
Morever, it can send emails from your website (server)
The link below is a tutorial for sending email through a contact form by means of PHP. Although it's old by roughly 3 years, the principles still stand.
http://apptools.com/phptools/forms/
PHP: Mail Functions
http://uk2.php.net/mail
PHP Mail
http://uk2.php.net/manual/en/function.mail.php
A word of warning, be sure to add mail headers as you dynamically generate the email to yourselves, otherwise, your email might be blocked or go straight inside junk email folder.
Watch out for email injections
http://www.phpit.net/article/php-security-3-example-exploits/