Can my travel agency software chase a customer who has not paid the balance for their trip?

Yes, it can. When you take a part payment you can record the amount still due and the date you need it by, and the software will send the customer a polite reminder on its own, so nobody on your team has to remember it.

The reminders go out by email under your own company name, and on some plans you can also send them by WhatsApp or text message. You choose how many days before the due date the first reminder goes, and whether a second one follows if the customer still has not paid. If the customer pays by card through your own booking page, the balance is marked as settled and the reminders stop by themselves.

There are two limits to keep in mind. The software only knows what has been entered into it, so if a customer sends money straight to your bank and nobody marks the payment as received, the reminder will keep going out and the customer will be annoyed. The other point is that your payment due date is your own, not the supplier's. An airline or a hotel may have its own ticketing or cancellation deadline that falls earlier, and the reminder to your customer has no effect on that, so you still need to watch the supplier date separately.

A good way to start is to set a balance due date on your next booking, send the first reminder to your own address as a test, and check that the wording and your company details read the way you want them to.