If anybody else is having a similar issue, here is how I fixed it:
$args = array(
'status' => 'processing',
'customer_note' => 'free sample',
'created_via' => 'sample request'
);
$order = wc_create_order($args);
$deliveryDateSelector = Date('j F, Y', strtotime('+3 days'));
update_post_meta($order->id,'Delivery Date', $deliveryDateSelector);
So far, so good.