Can WCart notify my own system when a new order comes in?

Yes, with webhooks. Open Plugins, then Webhooks in the plugin sidebar. It is marked Beta.

1. Under Add a webhook, enter the URL on your system that should receive the events.
2. Pick the events you want, for example order.created, order.shipped, order.cancelled or customer.registered. Reward point events are also available.
3. Click Create webhook. Copy the signing secret straight away, because it is shown only once. Your system uses it to check the X-WCart-Signature header on each call.

Subscriptions lists your webhooks, with Pause, Resume and Delete. Recent deliveries shows each event's status, attempt count and any error. If your system is down, WCart retries a failed delivery (up to 6 attempts in total), waiting longer each time.