I think you want to change the code on line 33 of class-wp-sponsors-widget.php to:
if(!($instance['category'] == 'all' || $instance['category'] == '')), since a blank category is what is getting passed for me from the widget settings.
I’m having the same issue, latest wordpress, latest plugin. I installed, added one sponsor, added widget with default settings, and nothing is showing up.
I debugged the query and interestingly enough, the query it generates:
SELECT wp_posts.* FROM wp_posts WHERE 1=1 AND (
0 = 1
) AND wp_posts.post_type = 'sponsor' AND ((wp_posts.post_status = 'publish')) GROUP BY wp_posts.ID ORDER BY wp_posts.menu_order ASC
Has a where clause that will always fail “0=1”.