This is totally doable. Most website builders today can either book appointments automatically or connect with a third-party service that does. Collecting payment is pretty much standard on most website builders now. Weebly is our recommended builder, and they have appointment booking forms right on their builder, so you can just drag-and-drop. They integrate with Paypal, Square, Authorize.net, and Stripe for payment processing. Hope that helps!
Let us know if you need anything else.