You can set your booking widget to private bookings, so that only the clients on your client list will be able to book appointments.
Set up private bookings
You can set up private bookings for your shop settings, by navigating to Settings > App Settings > Private Booking. You can read more about private bookings in this support article: Accept bookings from existing clients only.
Set up private bookings for a specific provider
You can also turn on private bookings for a specific provider, and still allow new clients to book with other providers in your shop.
To learn more about setting up a provider's private client list, check out this support article: Set a provider to private clients only.