EDIT

    '#default_value' => $link_template->category,

    '#autocomplete_path' => 'ext_links/category/autocomplete',

  $items['ext_links/category/autocomplete'] = [

    'title' => 'Autocomplete for external link categories',

    'page callback' => 'ext_links_category_autocomplete',

    'access arguments' => ['access administration pages'],

    'type' => MENU_CALLBACK

  ];

/**

 * Autocomplete function called via the menu hook ext_links/category/autocomplete

 *

 * @param $query_string

 *  the string to search for

 */

function ext_links_category_autocomplete($query_string){

  $matches = array();

  $sql = 'SELECT * FROM {ext_links} WHERE category LIKE :category';

  $args = [':category' => '%' . db_like($query_string) . '%'];

  $result = db_query($sql, $args);

  foreach ($result as $row) {

    $matches[$row->category] = check_plain($row->category);

  }

  // Return the result to the form in json

  drupal_json_output($matches);

}