Sure! Use the user_register filter (see here https://codex.wordpress.org/Plugin_API/Action_Reference/user_register ). You can call a custom function that will ‘fire’ after the user is added to the database. Your function would send out an email, perhaps.
Place the filter in your Child Theme’s function.php
file (you want to use a Child Theme so that your changes don’t get overwritten/killed by a theme update).
You could place it in your own special-purpose plugin – a simple plugin without settings that just runs. Put the filter in your plugin, and your done! (A special-purpose plugin is good for all sorts of things you want to do to your site, and doesn’t require modifying the functions.php file.) Learn about developing plugins here https://developer.wordpress.org/plugins/intro/ .