I set up the Webhooks integration under Integrations, but it is not offered as an option when I try to create a Metric alert. It is working as expected in Issue Alerts.

It looks like you are trying to configure webhooks using a legacy integration. For multiple projects, we recommend setting up an Internal Integration.

More info and example code snippets can be found here. The webhook should be available to use with metric alerts, so long as the "Alert Rule Action" option is toggled on in the Integration settings.