The reason this plugin no longer works has to do with Instagram. In April of 2018, they updated their api in two ways that causes this plugin to break. The first is that they have lowered the allowed number of api calls from 5000/hour to 200/hour. The second (and more important) is that they have depreciated the api url that the plugin uses to grab the information from Instagram.
If you open up the main plugin file in /wp-content/plugins/dsgnwrks-instagram-importer/dsgnwrks-instagram-importer.php and look on (or around) line 574, there should be line that creates the api_url variable. It should look something like the following:
$this->api_url = isset( $this->next_url ) && $this->next_url ? $this->next_url : $this->instagram_api . $this->settings->user_option( 'id' ) .'/self/media/recent?access_token='. $this->settings->user_option( 'access_token' ) .'&count=2';
If you replace that line with the following, the plugin should start working again.
$this->api_url = isset( $this->next_url ) && $this->next_url ? $this->next_url : 'https://api.instagram.com/v1/users/self/media/recent/?access_token='. $this->settings->user_option( 'access_token' );
I normally would advise against modifying a plugins code, but since this one hasn’t been updated in over a year, I think it will probably be okay. If they do update it, it will most likely include a fix for this, but in the meantime, this should work.
If you’re still having problems, you might need go into the plugin settings and change the date in which it is starting to pull from to one that is closer to todays date. The 200/hr request limit is most likely causing the plugin to fail. If you might also be hitting the limit if you have multiple version of the site (such as a testing server, live server, and any local development version of the site) hooked up to the same Instagram account and they are all set to pull every hour.