display the search field as soon as a page loads

Hey ae2agence44,

Thank you for the inquiry.

You can add this script in the functions.php file to automatically open the AJAX search on page load.

// custom scripts
function ava_custom_script_script()
{
    ?>
    <script type="text/javascript">
        (function($) {
            // open search bar on load
            $(document).ready(function() {
                $('#top #menu-item-search a').trigger('click');
            });
        })(jQuery);
    </script>
    <?php
}
add_action( 'wp_footer', 'ava_custom_script_script', 9999 );

Please make sure to purge the cache and do a hard refresh before testing.

Best regards,
Ismael