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I am considering rebuilding the store just so I can use your app builder and plan to create it using WordPress. Could you recommend a theme or plugin that you know of that has REST API?
Hello! WordPress has API http://developer.wordpress.com/docs/api/ and plugins - for example http://wordpress.org/plugins/json-api/ (you can search for more plugins online)
And so I can build a Wordpress site with the correct plugins and API and still develop the app with a barcode scanner to scan SKU/UPC ?
Yes, you can develop your site in WordPress. Add plugins you need and use this site as back-end for building your app with Appery.io.