Conceptually, the important thing is each stage waits to "ACK" the message until it's durably persisted. And when the message is sent to the next stage, the previous stage _waits for an ACK_ before assuming the handoff was successful.
In the case that your application code is down, the other party should detect that ("Oh, my webhook request returned a 502") and handle it appropriately -- e.g. by pausing their webhook queue and retrying the message until it succeeds, or putting it on a dead-letter queue, etc. Your app will be "out of sync" until it comes back online and the retries succeed, but it will eventually end up "in sync."
Of course, the issue with this approach is most webhook providers... don't do that (IME). It seems like webhooks are often viewed as a "best-effort" thing, where they send the HTTP request and if it doesn't work, then whatever. I'd be inclined to agree that kind of "throw it over the fence" webhook is not great and risks permanent desync. But there are situations where an async messaging flow is the right decision and believe it or not, it can work! :)