Hello Laura,

Thanks for reaching out to us. There is no option to block the customer from booking the appointment but with Amelia 7.0 we've added the option to mark the customers with a "no show" tag. 

Once you enable this option, all customers who had appointments marked as “No Show” in the past will be seen differently in the “Appointments” section, and the Customer’s profile (in the “Customers”) section. A gray tag indicates the customer who failed to show up once; a yellow tag indicates the customer who failed to show up twice; and the red tag indicates the customer who failed to show up 3 or more times:

You can also set the default status to pending, so every time that customer books an appointment, you can cancel it in the backend.

I hope it helps.