Hey Jeannette,
You could create something, but I think it would be quite tedious to create and you would not be able to automate links being greyed out as time slots were filled.
You could create a landing page with a form that utilizes the custom URL builder to enter times into a hidden field. Then, when they click submit, you can send them an automated email, but you would then need to go into the landing page to remove that time from the availability.
Overall, I think you could make it work, but it would be quite the project. How many dates would you have? One person/date?