I cannot provide the code, but you could do it with a custom plugin.
Listen for our actions, e.g. woocommerce_order_status_completed. This is fired when the order is complete. You can use this hook to loop over the items in the order and ping your external service using wp_remote_post.
https://codex.wordpress.org/Function_Reference/wp_remote_post
http://code.tutsplus.com/articles/the-beginners-guide-to-wordpress-actions-and-filters--wp-27373
Alternatively, for webhooks, see https://woothemes.github.io/woocommerce-rest-api-docs/#webhooks16
