You can remove the avatars from the donors widget by creating a custom template for the donors widget. To do that, create a new file in your theme or child theme at charitable/widgets/donors.php.

Next, copy and paste the contents of the charitable/templates/widgets/donors.php template file (this is in the Charitable plugin directory) into this file. You can then make changes to the version in your theme without touching the core code. To remove the avatar specifically, you just need to delete this line:

echo $donor->get_avatar();