Where both systems have an API, the integration listens for events (a webhook fires when an order is placed or an invoice is paid) and writes the change to the other side. A scheduled sweep runs behind it to catch anything the webhooks missed. Every write is keyed so a retry can't create a duplicate, and a failed write gets kept and replayed.
Where a system has no API, we work with what it does have. Scheduled file imports and exports, direct database access where the vendor permits it, parsing the emails it sends, or automating its own screens when there's nothing else. Those paths break more often than a real API, so we point more monitoring at them.
Either way, a reconciliation report runs on a schedule. It lists every record that exists on one side and not the other, and every one where the amount or the status disagrees. Integrations fail quietly. That report is what makes a quiet failure loud. All of it gets built against sandbox or test accounts first, and it only points at your live systems after you've signed off the mapping.