Appointment slots: Is there a way for the number of available reservations be tied to how big the party is?

Also, is there a way for the number of available reservations be tied to how big the party is? Like if someone reserves a time for a party of 4, can that make the available slots go down by 4?

If I understood you correctly you want to allow your users to reserve multiple tickets for same date and time. Unfortunately that is not possible to achieve with Appointment Slots widget. To reserve time for a party of 4 your users will need to submit your form 4 times.

However we have widgets that will allow your users to select more of same options. 

Inventory widget 

Quantity Gift Registry widget

Hope this will help. Let us know if you need further assistance.