This depends on which suppliers you work with and what your software plan includes. Some suppliers offer direct connections that let bookings flow automatically between your travel agency software and their systems, which means you enter the booking once and it updates everywhere. Other suppliers may not have this kind of connection available, or the connection might work only one way, so you would still need to manage some information manually or check both systems to make sure everything matches up.
When you set up your travel agency software, the suppliers you connect to will determine what is possible. A supplier with a modern API and integration options will give you more automatic syncing than one with older systems. Your software plan also matters, because some plans include more supplier connections and automations than others. The best approach is to check with your software provider which of your main suppliers can sync automatically and which ones cannot.
If automatic syncing is important to your workflow, ask your software provider for a list of suppliers that offer real-time or regular syncing, and whether you can add more supplier connections if you need them. This will help you decide whether the setup matches how you want to work and whether you will spend less time managing duplicate information across different systems.