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 outside 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.
Then, use the ability to customize email templates for each appointment type (click to read more) to make the confirmation email for the Request appointment type read as a "request received" email and the confirmation email for the Confirmed appointment type read as a confirmation email. Head to Email Settings to make that customization.
When someone books a request appointment type 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 type and click "Save Changes".
Once you've edited the appointment type, you can use the cog at the top right of the appointment details to resend the confirmation email. Because you have customized your email templates, the client will now get a confirmation email.
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.