In this situation we want a custom field on checkout which has a jQuery Date selector:

/**
 * enqueue jquery datepicker script
 *
 * If is not admin (backend) and the page is 
 * woocommerce checkout add the field
 */
 

 
add_action( 'wp_enqueue_scripts', 'puddinq_enabling_date_picker' );
function puddinq_enabling_date_picker() {

    // Only on front-end and checkout page
    if( is_admin() || ! is_checkout() ) return;

    // Load the datepicker jQuery-ui plugin script
    wp_enqueue_script( 'jquery-ui-datepicker' );
}

/**
 * Add the field to checkout
 *
 * echo a 
 with the script, styling and field
 */
 
add_action('woocommerce_after_order_notes', 'puddinq_delivery_date_selector', 1, 1);
function puddinq_delivery_date_selector($checkout) {
    echo '
' . __('Afleverdatum voorkeur') . '
';
    echo ' 
  ';
    echo '';

    woocommerce_form_field('puddinq_delivery_date', array(
        'type' => 'datetime',
        'class' => array( 'puddinq_delivery_date_class form-row-wide'),
        'label' => __('Date') ,
        'placeholder' => 'Select a date',
        'id' => 'datepicker'
    ),
    $checkout->get_value('puddinq_delivery_date'));

    echo '
';
}

/**
 * Add the field to checkout
 *
 * echo a 
 with the script, styling and field
 */
add_action('woocommerce_checkout_update_order_meta', 'puddinq_save_date_selector_meta');

function puddinq_save_date_selector_meta( $order_id ) {
    if ($_POST['puddinq_delivery_date']) update_post_meta( $order_id, 'puddinq_delivery_date', esc_attr($_POST['puddinq_delivery_date']));
}