First outside your shortcode callback function just register the stylsheet and then in shortcode callback function enqueue it. like:-

        function wpsoe_191512_avatar_shortcode_wp_enqueue_scripts() {
      wp_register_style( 'get-avatar-style', plugins_url( '/css/style.css' , __FILE__ ), array(), '1.0.0', all );
      }
    
    
      add_action( 'wp_enqueue_scripts', 'brw_avatar_shortcode_wp_enqueue_scripts' );
    
      if ( function_exists( 'get_avatar' ) ) {
    
      function wpse_165754_user_avatar_shortcode ( $attributes ) {
                global $current_user;
                get_currentuserinfo();
                extract(shortcode_atts(array(
                 "id" => $current_user->ID,
                 "size" => 32,
                 "default" => 'mystery',
                 "alt" => '',
                 "class" => '',
          ), 
          $attributes, 'get_avatar' ));
          $get_avatar= get_avatar( $id, $size, $default, $alt );
    
          wp_enqueue_style( 'get-avatar-style' );
    
          return '<span class="get_avatar '.$class.'">'.$get_avatar.'</span>';
          }
          add_shortcode ('get_avatar', 'wpse_165754_user_avatar_shortcode');
          }

Above is the example. You can use yours.
Thanks