Add a term under a vocabulary when a content of particulr type is added
In short, "yes".

Using hook_node_insert() where you can assign a category term value programmatically. Note that this is an over-ride (hook) of the node creation process. Basically do a Googe search for Drupal 7 node creation programmatically", example: Guide to programmatic node creation in Drupal 7. Look for the details on "Term reference (taxonomy) field".

Note that this an advanced Drupal task that involves creating a custom module that's specific to your site. If you're not already familiar with this process of applying hook functions to alter default Drupal functionality then I suggest you do some research on basic Drupal requirements for custom module creation first.

Good luck and keep asking questions, you'll figure it out and might become a Drupal rock star in the process!