I am unable to register a webhook.
I do go though the oAuth flow, which completes successfully. But when trying to trigger the webhook creation endpoint, just like in the example I do end up with an 403 error where the response body does not contain any data that would give more information to why is this happening.

The documentation does not state any thing about this behaviour, or if it requires some additional configurations.
Doc reference: https://developer.zendesk.com/api-reference/webhooks/webhooks-api/webhooks/#create-or-clone-webhook
Are you using an OAuth token to authenticate your request to create the webhook? If so, did you set the scope to give write access to all resources or write access to webhooks?