I want to make sure that form will only send submission to my client.

for my client intake form, I want to make sure that from my Squarespace website the form will only be sent to the client after they have booked an appointment. Is that correct?

You need to configure the notification to send only email to your client's email address, here's the guide: How-to-Change-the-Email-Address-Used-for-Notification-Email

Let us know if you are referring to something else.