edit_comment_link is not showing for comment author

First of all, you don’t want unregistered users to be able to edit comments, because you have no way of knowing whether they are who they say they are. So, this can only be made to work for registered users.

As you can see from roles and capabilities only editors and up can moderate comments. You can give this capability to others, however, using code like this:

Update Using edit_comment is better than moderate_comments, which would be too broad for your purpose, I guess.

function wpse250786_add_cap() {
        $role = get_role( 'contributor' );
        $role->add_cap( 'edit_comment' );
        }
add_action( 'admin_init', 'wpse250786_add_cap');