Using osTicket's cron API
We ware struggling getting the HTTP API cron task to run. We found that (at the time of writing) the official documentation doesn't work for our version: 1.15.8. After many tries we found the correct way to call the cron API:
curl -d "{}" -H "X-API-Key: API_KEY_HERE " https://YOUR_DOMAIN/api/tasks/cron
In short, it needs to be a POST request, that has the X-API-Key header. Obviously you'll need to generate the API key for the IP you're executing the request from.
I hope I saved you some time.