How to add autocomplete to custom taxonomy for CPT

The logic in the tax query is very unlikely to verify true.

Look at it:

'tax_query' => array(
    'relation' => 'AND',
                  array(
                     'taxonomy' => 'zip_code',
                     'field'    => 'slug',
                     'terms'    => array( $term ),
                  ),
                  array(
                     'taxonomy' => 'city_served',
                     'field'    => 'slug',
                     'terms'    => array( $term ),
                  ),
),

If you read the tax query, it says: get posts that have assigned the same value ($term) in zip_code and city_served taxonomies.

Note that you have set the AND relation between both taxonomies and use the same $term variable for both. It is very unlikely a zip code and a city name has the same value.

Maybe you want this other logic:

'tax_query' => array(
    'relation' => 'OR',
                  array(
                     'taxonomy' => 'zip_code',
                     'field'    => 'slug',
                     'terms'    => array( $term ),
                  ),
                  array(
                     'taxonomy' => 'city_served',
                     'field'    => 'slug',
                     'terms'    => array( $term ),
                  ),
),

Apart of that I don’t see anything wrong in your code, although you are not showing the full code, so I can not been sure.

I’ve built a simple test using default post type, category and tag taxonomies ant it works:

PHP:

add_action( 'wp_ajax_hbgr_search', 'hbgr_search' );
add_action( 'wp_ajax_nopriv_hbgr_search', 'hbgr_search' );
function hbgr_search() {
        $term = strtolower( $_GET['term'] );
        $suggestions = array();

        $input_args = array(
            'post_type' => 'post',
            'tax_query' => array(
                'relation' => 'OR',
                array(
                    'taxonomy' => 'category',
                    'field'    => 'slug',
                    'terms'    => array( $term )
                ),
                array(
                    'taxonomy' => 'post_tag',
                    'field'    => 'slug',
                    'terms'    => array( $term )
                ),
            ),
        );

        $loop = new WP_Query( $input_args);

        while( $loop->have_posts() ) {
            $loop->the_post();
            $suggestion = array();
            $suggestion['label'] = get_the_title();
            $suggestion['link'] = get_permalink();

            $suggestions[] = $suggestion;
        }

        wp_reset_postdata();

        $response = wp_send_json( $suggestions );

}

add_action( 'wp_enqueue_scripts', 'add_scripts' );
function add_scripts() {
    wp_enqueue_script( 'jquery' );
    wp_enqueue_script( 'jquery-ui-autocomplete' );
    wp_register_style( 'jquery-ui-styles','http://ajax.googleapis.com/ajax/libs/jqueryui/1.8/themes/base/jquery-ui.css' );
    wp_enqueue_style( 'jquery-ui-styles' );
    wp_register_script( 'my-autocomplete', plugin_dir_url( __FILE__ ) . 'my-autocomplete.js', array( 'jquery', 'jquery-ui-autocomplete' ), '1.0', false );
    wp_localize_script( 'my-autocomplete', 'MyAutocomplete', array( 'url' => admin_url( 'admin-ajax.php' ) ) );
    wp_enqueue_script( 'my-autocomplete' );
}

JavaScript (my-autocomplete.js):

(function( $ ) {
    $(function() {
        var url = MyAutocomplete.url + "?action=hbgr_search";
        $( "#search-dealer" ).autocomplete({
            source: url,
            delay: 300,
            minLength: 3
        });
    });

})( jQuery );

Search form:

<form method="get" action="<?php echo esc_url( home_url( "https://wordpress.stackexchange.com/" ) ); ?>" role="search">
    <input id="search-dealer" type="search" placeholder="Search" name="s">
    <button type="submit">Submit</button>
</form>