I had this issue as well, the problem is that everyone using the plugin is using the same client id. Likely something changed where a limit was hit, or some policy was violated, or an agreement/API was changed, so that client id doesn’t work anymore.
I was able to fix it by going to https://instagram.com/developer/clients/register/ and following the steps until I got my own client id. I then went to my blog, Plugins->Editor->Select Basic Instagram Widget->Edit “basic-instagram-widget/basic-instagram-widget.php”
Find:
“if ( ! defined( ‘BIW_CLIENT_ID’ ) ) {“
on the next line you’ll see:
define( ‘BIW_CLIENT_ID’, ‘somecode’ );
Replace the somecode with your own client id. Click “Update File” and then you’re done!