Drupal form creating method without using a module
As the comments and the other answer says, there are several possibilities:


A contributed module such as Webform.
Someone with very bad habits hard-coded that form inside templates, other modules or even Drupal core.
A block/node could contain that form, specially if the PHP-filter is enabled. Even with the HTML filter enabled (and submitting the form to another page).


If you see some custom text, or the name of the fields are unique or rare, you could try to search inside Drupal root directory for them:

grep -r 'Custom text' /var/www/drupal


phpMyAdmin includes a functionality to search across an entire database. If you feel desperate, try to search inside the Drupal database for the text you see in the form.

Examining the outputted HTML code could help. You will see if the form has an ID, for example.