There are several ways to allow people to book a time slot and then pay for it. I can write and code it myself, and have done this before, but then you are dependent on me for any major changes you might require in the future. Good flexible functionality can be purchased as a "plug-in" which allows for flexibility and multiple payment methods. There are very good ones that currently cost about $50 a year. Free ones that I have seen in the past, have all begun to charge. So before making the yearly commitment, some people will begin accepting bookings and payments over the phone (to the business office number) and then manually enter those purchased time slots in a scheduler or calendar of their choosing.
For this site, I prefer to use Amelia, one of the most popular online booking and payment plug-ins for a WordPress site. (WordPress is more easily maintainable when changes are required in the future. Plug-in makes it sound very simple, but it will still require some instructions on how to update costs, how to allow for multi-hour slots, how to set up a limit on the number of people allowed to book a slot. How to set up slots that start on the half-hour instead of just the hour, etc.
The payment goes through the "Cart" or a link to any of several payment sites and is easy to pay online. You will have to enter bank account info for any payment method like Stripe, Venmo, Square, PayPal, etc.
There are methods like making each time slot a separate product that is purchased and moved into the CART. This sound intuitive but requires hundreds of products per year.
Another method is to create event tickets, but has the similar problem of setting up multiple tickets in advance.
Amelia basically allows a simpler way to pay in advance for an appointment. You may have to decide later how flexible you may have to be to allow slots to be booked before payment is received, and Amelia allows for you to set up something as pending until the payment is received (in case you allow persons to bring a credit card into the physical area rather than pay in advance).
