Intergration with booking calendar

We are a kindergarten. We want to create an admission registration form integrated with booking/ appointment calendar, for customers who did not visit our facility before. Thus, when they register through our website, we would like to ask them if they have visited already (conditional logic in the form); if they say “yes”, then they proceed with standard registration. However, if they say they have not visited before, a booking calendar needs to open and they can select visit date. Is this possible?

Further, which booking/ appointment plugin is compatible with this form. Our visit dates are pre-defined, so customers should only be able to select from pre-defined dates (and predefined time-slots) until slots are fully booked.