You cannot make a contact form without PHP code that sends the email. If you do not use a plugin, you have to write the code yourself and put it somewhere else, like in a theme. But putting it in a plugin is better. You also cannot put PHP code in the page or post (without a plugin for that).
The PHP function that WordPress provides for sending an email wp_mail() is for use in a plugin or theme (not in a page or post, since you cannot add PHP there).