Subscribers can’t seem to be able to add new items to the taxonomy list. Only Admins can add new items.
Why is this the case? Can I open the permissions so that they can?
Here are my full settings for the field.
This is a WP thing. What uses can do with this field is decided by their role and permissions in WP. To allow subscribers access to this you need to look into something like https://wordpress.org/plugins/user-role-editor/ and give subscribers the needed permissions.