Para incluir el selector de Fecha y Hora basta con insertar este código en el archivo functions.php de tu tema hijo (recomendado) o el activo.

// Register main datepicker jQuery plugin script
add_action( 'wp_enqueue_scripts', 'enabling_date_picker' );
function 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' );
}

// Call datepicker functionality in your custom text field
add_action('woocommerce_before_order_notes', 'my_custom_checkout_field', 10, 1);
function my_custom_checkout_field( $checkout ) {

    date_default_timezone_set('America/Los_Angeles');

    ?>
    <div id="my_custom_checkout_field">
    <h3><?php _e('Check In Date'); ?><abbr class="required" title="required">*</abbr></h3>
    <?php

   woocommerce_form_field( 'order_in_date', array(
        'type'          => 'text',
        'class'         => array('form-row-first'),
        'id'            => 'from',
        'required'      => true,
        'placeholder'       => __('Select Date in'),
    ),$checkout->get_value( 'order_in_date' ));

    woocommerce_form_field( 'order_out_date', array(
        'type'          => 'text',
        'class'         => array('form-row-last'),
        'id'            => 'to',
        'required'      => true,
        'placeholder'       => __('Select Date out'),
        'clear'         => true
    ),$checkout->get_value( 'order_out_date' ));

    ?>
    </div>

    <script>
        jQuery(function($){
            var dateFormat = "yy/mm/dd",
                from = $( "#from" ).datepicker().on( "change", function() {
                      to.datepicker( "option", "minDate", getDate( this ) );
                }),
                to = $( "#to" ).datepicker().on( "change", function() {
                    from.datepicker( "option", "maxDate", getDate( this ) );
                });

            function getDate( element ) {
                var date;
                try {
                    date = $.datepicker.parseDate( dateFormat, element.value );
                } catch( error ) {
                    date = null;
                }
                return date;
            }
        });
    </script>
    <?php
}

En el calendario tiene activa la opción para pasar entre los diferentes meses