Hey Geoff,
We've created several apps for both iPhone and Android that use CMS Builder as the back end to inject content into the App.
We use phoneGap build, which is a system created by Adobe that allows you to turn a site into an app. The app is essentially a browser that opens the site for you. You can create apps for several different devices, including iPhone and Android.
You can find out more on this system here: https://build.phonegap.com/
We've found that Google is fairly lenient in what they'll add to the Play Store. But the Apple app verification process is much stricter, they insist that the app isn't just a website wrapped in an app and that it has some app-specific functionality (for example a QR code scanner, push notifications, etc).
Let me know if you have any questions.
Cheers,
Greg
Greg Thomas
PHP Programmer - interactivetools.com