Slide out booking system custom code

Site URL: https://www.quay.com.au/

Hi my client is using a booking system, the same company the business in the example url is using. Is there a plugin or any custom code available to make a slide out effect with squarespace? I have access to the booking systems account/embed code details. But they don't offer that effect with their widgets. "Nowbookit" is the company who i spoke with and they confirmed Quay restaurant (example url) are using custom code to make the slide out effect. 
Thank you