You have 2 options to enable this video calling right from Shopify store. One way is via Facebook messenger by integrating the SDK inside. You can have auto-reply to let them call you on Zoom or Skype.
Another option is to have using Intercom along with Aircall for the customer to directly call you from the website - https://www.intercom.com/app-store/?app_package_code=aircall-now&category=phone-and-video
We created the Instaconnect app to add 1-1 live video chat and guided webinars to Shopify. Webinar content is automatically generated from product descriptions and photos, or you can upload your own content.
You can try it for free here:
Hope it helps - let me know if you have any questions and if it works out for you?
Try Consolto.com's videochat straight out of your website.
The solution has not only videochat but also screen-sharing, text-messaging, file-sharing, audio-recording, customer insights and even meeting-scheduling.
The Shopify app can be found here: https://apps.shopify.com/consolto-videochat
The app you are looking for is called UPTOK. Its a Video chat for online shops. And it offers very unique features for Shopify stores. You can look up products and suggest to shoppers during sessions. You can also offer personalized discount codes to shoppers live.