It is possible to add a Book.app shortcut icon to your iPhone or iPad home screen so that it displays and opens like an app
To do so, first ensure you have added an Online Icon Image to your mini-site custom settings. The image you add here will be the app icon on your device's home screen.
Go to the Online Booking > Book.app > Customise Site page
In the Upload your logo section, click on the Select File... button next to Online Icon Image, then follow the prompts to add your image
See How do I customise my Book.app mini-site? for more information on customising your mini-site
Then open the Safari browser on your iPhone or iPad and go to your Book.app mini-site
You can locate your mini-site URL via the web app by clicking Online Booking > Book.app > Visit your Site. The URL will display in the browser bar.
When you have your Book.app mini-site open in Safari on your device, tap the Action icon in the bottom tool bar
Scroll up through the list of options and tap Add to Home Screen
The Online Icon Image will display on your home screen the same as your other apps. Tap on it to open your mini-site like an app.