The live system should 'push' the updates to the website. How they do that depends on what platform the live system is running on.
The usual problem you will encounter is that the people who develop or maintain that system won't cooperate, won't have a budget for it, and frequently don't have the neccessary skills to implement a robust system, or produce compatible output.
If they are not locked into some legacy system that will take 6 months and cost a fortune to modify then see if they know how to produce an RSS feed in XML that your site would use to auto-update itself. That will future-proof things and have notable benefits down-stream. An RSS feed could, for instance, be made available to agents and customers to keep them up to date on their existing bookings.
I mostly do back office systems and frequently find that the corporate MIS will not co-operate with the business unit I am dealing with. We end up working around them and do things like screen scrape a terminal emulator running under one of the business depts user accounts. You just set it up to continually cycle through the system looking for update to capture. Totally inefficient - but it works.