How do I send POST to external API from WooCommerce

I am building an eCommerce site for a client. I have done eCommerce before, so most of this is nothing new to me. However, this client has an in-house order management program they need WooCommerce to integrate with because they also take call-in orders. Basically, we want to use WooCommerce for the storefront, but any time an order is created, it should send all order details to their system. They also want to be able to use their own payment gateway so that they can manage refunds and returns from their end.

Trouble is, this type of integration is quite new to me (I was upfront with them about this, so they know it’s a learning curve for me), so I’m not even really sure where to start. I have found lots of documentation for how to modify the WooCommerce shop via API from an external system, but not how to send API POSTs from WooCommerce to an external API. I’m a quick study, though, so if anyone can help point me in the right direction for documentation on how to do this, it would be most appreciated. I’d also be interested in existing WooCommerce extensions that would make this easier, if such exists.