Acuity Scheduling is designed to make online scheduling as hands-off as possible, which is why we do not offer an option to request appointments rather than schedule them. Instead, availability in Acuity is powerful enough that you can set the exact times you're available, including adjusting availability for one-off days with longer or shorter hours, and syncing your Acuity calendar(s) with your Google calendar to block off any existing events.
If this is critical for your business, however, there is a workaround!
This workaround hinges on you setting up TWO appointment types, one public and one private. The public one will be the one to show on your scheduling page. You could call it something like "REQUEST: Pet Instagram Photo Shoot".
Then you'd create a duplicate appointment, name it "CONFIRMATION: Pet Instagram Photo Shoot", and set it to private.
When someone sends in a request that you want to confirm, you would select the appointment from the Appointment Calendar, click "Edit" and change the appointment type from the REQUEST appointment to the CONFIRMATION one and click "Save Changes".
Once you've edited the appointment type, you can then Reschedule the appointment to the same date and time by clicking on the button that says "Custom". Rescheduling will will send the Reschedule e-mail, which you can customize to be more of a confirmation e-mail, if you'd like (head over to your Email Settings to customize!).
Here's a handy screen recording of those steps!
You can can also put a disclaimer in your Scheduling Instructions indicating that clients are sending through a request, and that you’ll follow up within 24 hours to confirm or cancel. You can change your appointment confirmation notification to include a similar sentiment as well, and you can even change your scheduler's wording, such as changing the title of the Confirmation page to something more along the lines of Appointment Request.