How can I make a form that would allow clients to choose appointment times and pay for sessions?

Is there a way that I can have available times that are set by me and then when they are booked another client cannot choose? 

To be able to allow the user to choose a date and limit the date, you may be able to use our Date Reservation widget:

https://widgets.jotform.com/widget/date_reservation

Related widgets:

https://widgets.jotform.com/widget/gift_registry

https://widgets.jotform.com/widget/appointment_slots

Let us know if you have further questions.

Thanks.