In WooCommerce, I can get a field hooked into my WooCommerce cart page & showing up OK using the following in my theme's (Storefront) functions PHP:
<?
            // ADD Custom Fields to Checkout Page
            /**
             * Add the field to the checkout
             **/
            add_action('woocommerce_after_order_notes', 'my_custom_checkout_field');
            function my_custom_checkout_field( $checkout ) {
                date_default_timezone_set('America/Los_Angeles');
                $mydateoptions = array('' => __('Select PickupDate', 'woocommerce' )); 
                echo '<div id="my_custom_checkout_field"><h3>'.__('Delivery Info').'</h3>';
               woocommerce_form_field( 'order_pickup_date', array(
                    'type'          => 'text',
                    'class'         => array('my-field-class form-row-wide'),
                    'id'            => 'datepicker',
                    'required'      => true,
                    'label'         => __('Delivery Date'),
                    'placeholder'       => __('Select Date'),
                    'options'     =>   $mydateoptions
                    ),$checkout->get_value( 'order_pickup_date' ));
                echo '</div>';
            }
            /**
             * Process the checkout
             **/
            add_action('woocommerce_checkout_process', 'my_custom_checkout_field_process');
            function my_custom_checkout_field_process() {
                global $woocommerce;
                // Check if set, if its not set add an error.
                if (!$_POST['order_pickup_date'])
                     wc_add_notice( '<strong>PickupDate</strong> ' . __( 'is a required field.', 'woocommerce' ), 'error' );
            }
            /**
             * Update the order meta with field value
             **/
            add_action('woocommerce_checkout_update_order_meta', 'my_custom_checkout_field_update_order_meta');
            function my_custom_checkout_field_update_order_meta( $order_id ) {
                if ($_POST['order_pickup_date']) update_post_meta( $order_id, 'PickupDate', esc_attr($_POST['order_pickup_date']));
            }
            ?>
However, the datepicker widget is not initiated when you click into the field. I know the following code is required in the header for JQuery to work:
<script>
$( function() {
$( "#datepicker" ).datepicker(); } );
</script>
This did not work for me, so I thought to put this function into the checkout.js file in Storefront theme, but the added field didn't have calendar widget functionality.
There's a lot of .js in the them do I need to start a new one for includes?
First you will need to:
jQuery(function($){ instead of $(function(){ …So to enable datepicker for your custom text field your code will be:
// 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_after_order_notes', 'my_custom_checkout_field', 10, 1);
function my_custom_checkout_field( $checkout ) {
    date_default_timezone_set('America/Los_Angeles');
    $mydateoptions = array('' => __('Select PickupDate', 'woocommerce' ));
    echo '<div id="my_custom_checkout_field">
    <h3>'.__('Delivery Info').'</h3>';
    // YOUR SCRIPT HERE BELOW 
    echo '
    <script>
        jQuery(function($){
            $("#datepicker").datepicker();
        });
    </script>';
   woocommerce_form_field( 'order_pickup_date', array(
        'type'          => 'text',
        'class'         => array('my-field-class form-row-wide'),
        'id'            => 'datepicker',
        'required'      => true,
        'label'         => __('Delivery Date'),
        'placeholder'       => __('Select Date'),
        'options'     =>   $mydateoptions
        ),$checkout->get_value( 'order_pickup_date' ));
    echo '</div>';
}
Code goes in function.php file of your active child theme (or theme) or also in any plugin file.
Tested and works. On storefront theme you will get that:

You may need to style it…
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