How to display featured images for all posts with certain taxonomy label?

Something quite rough that needs improvement but can give you a starting point, if I’ve understood your needs: use [dr] as shortcode marker where you want

function my_dream_shortcode($atts, $content = null) {
  ob_start(); ?>
  <ul>
    <li class="button" onClick="get_data('corporate')">Corporate</li>
    <li class="button" onClick="get_data('sales')">Sales</li>
    <li class="button" onClick="get_data('support')">Support</li>
  </ul>
<div class="my_plugin_result"></div>
<style>
  li.button{
   list-style:none;
   padding:4px 10px; 
   background-color:cadetblue;
   margin:10px;
   float:left;
   min-width: 160px;
   text-align: center;
   cursor:pointer;
  }
  .my_plugin_result figure{
   float:left;
   padding:4px;
   background:#ccc;
}
</style>
<script>
  var myPluginAjaxUrl="<?php echo admin_url( "admin-ajax.php'); ?>';
  function get_data(term) {
    jQuery.ajax({
      url: myPluginAjaxUrl,
      data: {
        action: 'get_data_for_my_shortcode',
        term: term
      }
    }).done(function (response) {
      console.log(response)
      jQuery(".my_plugin_result").html(response);
    });
  }
</script>

<?php
 return ob_get_clean();
}
add_shortcode("dr", "my_dream_shortcode");

add_action('wp_ajax_nopriv_get_data_for_my_shortcode', 'get_data_for_my_shortcode');
//add_action('wp_ajax_get_data_for_my_shortcode', 'get_data_for_my_shortcode');
function get_data_for_my_shortcode(){
  global $wpdb;
  $args = array(
   'post_type' => 'post',
   'tax_query' => array(
      array(
        'taxonomy' => 'leadership',
        'field' => 'slug',
        'terms' => $_REQUEST['term']
      )
    )
 );
  $response="";
  $query = new WP_Query( $args );

  while($query->have_posts() ):
    if($query->have_posts() ):
      $query->the_post();
      $response.='<figure><img src="'.get_the_post_thumbnail_url(get_the_ID(),'medium').'"/></figure>';
    endif;
  endwhile;
  echo $response;
  die();
}

Thanks to @bulldog edited code to work effectively in front-end