Solution:
You can use the WooCommerce REST API v2 or later to get the Order Meta Data which includes these custom fields. If you GET an Order:
wp-json/wc/v3/orders/3454
it will include a meta_data array with these custom field values:
"meta_data" :
[
{
"id" : 4672,
"key" : "_shipping_method",
"value" : [ "flat_rate:1" ]
},
{
"id" : 4673,
"key" : "_shipping_phone",
"value" : "08 9632 7412"
}
]