Can a WordPress site be converted into an iOS app? Maybe using something like PhoneGap to encapsulate the php?
Most definitely! There is a WordPress plugin called AppPresser which connects WordPress to PhoneGap, making it easy to access native device features and create an application that feels less like a wrapper. One of the challenges mentioned above is offline access, however, that is currently being worked on. Currently, there are a number of extensions to enable native device features as well as adding e-commerce functionality via Woocommerce.
AppPresser requires an app-specific theme to work (you can always build your own) but fortunately, there is the AppTheme provided by the makers of AppPresser, which offers an optimized app-style theme that is more likely to be approved by Apple's review team.
You can see AppPresser in action (there are a few apps available now using it) by checking out their website: http://apppresser.com.