You need the API up; that's how it works.

However, should you fail to renew a cert, your script should just keep retrying until the API is back and you should allow enough time for it to fail. I believe most clients renew certs ~10 days in advance.

Obviously if you're relying on getting a new cert to bring up something like a preview environment or to hand out your own subdomains, this will result in a downtime/delay in provisioning, but most people would be fine with a single wildcard and never really experience a problem, as long as their script runs and they keep it reasonably up-to-date.