Yes, in most cases you can. Your booking history and your customer records belong to you, and travel agency software is normally able to produce a download of them in a spreadsheet file that you can keep safely or hand over to a new system later on.
What comes out usually covers the information you use day to day, such as passenger names, travel dates, destination, the supplier reference, the booking status and what was paid. Some things travel less easily, for example the original supplier confirmation documents or free text notes your staff typed against a booking, because every system stores those in its own way. Exactly which fields are included, and whether you can download them yourself or need to ask the support team to prepare the file, can depend on your plan, so it is fair to ask before you assume.
It also helps to remember that getting the information out is only half the job. The system you move to has its own fields and its own rules, so expect some manual tidying, particularly with older records. And if a booking was made through a supplier connection, the live record still sits with that supplier, which means your download is a copy of your history rather than something a new system can act on or amend.
Before you commit to any software, ask to see a sample download file so you know what you would actually get, check your own data protection duties for holding customer details, and take a copy now rather than waiting until the day you want to switch.