The BigCartel developers v1 API states:
Webhooks will be sent as an HTTP POST request to the webhook_url for your application
Where is this webhook_url set?
The Big Cartel API is currently in Beta, and in order to develop an application that uses webhooks you'll have to apply for an invite. More info on that is here: https://help.bigcartel.com/developers/api/v1/#public-projects