Can a workflow start when an appointment is booked?

Yes. Workflows have an Appointment scheduled trigger and several other appointment related triggers. Check the trigger list in the workflow builder.

A common use is stopping a follow-up sequence once someone books:

1. In the same agent that runs the sequence, create a second workflow.
2. Set the trigger to Appointment scheduled and select the calendar.
3. Add a step that deactivates the contact from all workflows in this agent.

After the person books, they stop getting the sequence messages.