Can a WordPress site be converted into an iOS app?

Be careful when simply wrapping things into a mobile app. This can backfire on you and you could be presented with some very harsh edge cases. PHP isn't going to be encapsulated so much as the HTML/CSS/JavaScript. So keep in mind your mobile application could likely be in a position of requiring an online connection to work (because it must interface with your existing hosted WordPress site). This could also mean your hosting solution needs to be evaluated to ensure you can handle any increased traffic (and those traffic patterns could be different when coming from a mobile app that perhaps loads things the user doesn't see right away, accesses content that may not be cached, etc.). You want to ensure your server doesn't go down because then your mobile app would be "down" as well.

That said... Things like Phonegap (web views) are a wonderful idea for utility apps because the performance is good enough for those (and hey even some games) and they end up being easily ported across many mobile operating systems.

I would look into Phonegap, Appgyver (a new and totally awesome one because parts of it will utilize the native OS and your app will feel more responsive), Appcelerator Titanium, and perhaps even the new Famo.us one. Also take a look at the Ionic Framework for some further ideas about mobile UI and what you can do with these web view style apps.

I'm a developer at Clarity and we use PhoneGap to publish our native iOS application to the App Store.

From experience, wrapping our website in PhoneGap has been a really good way to get a native application that has native features like push notifications for "cheap".

The nature of Clarity itself requires you to be online to use the product: browse and search, request calls, ask questions, etc. For that reason, we never looked at providing offline support. If we can't find an internet connection, we show a very simple alert dialog.

Another concern that we had is startup performance, especially on subsequent starts. To remedy that we added a simple caching layer to the native code that allows us to save our biggest assets (JavaScript and CSS) locally and avoid having to re-fetch them unless they changed locally.

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.

No you can't encapsulate it.
You can crept APIs for your content or implement a theme that is responsive and looks good on mobile web but there is no way of directly wrapping your existing code.
Also, a part from the technical side, Apple always tries to reject apps that are purely copies of websites.
Please make good apps, that do something.

We've actually accomplished this with the Marketing Technology Blog through a third-party platform that optimizes the visual content and lays out the posts and pages in an application that's beautiful but never requires updating. We did this with a company called Postano who have several clients on it. We're working on actually partnering with them to deploy it to more people. The platform they built connects to your WordPress site but outputs native applications for both Android and iOS (and others). It's not using PHP OR WordPress - just driving and optimizing the content FROM WordPress to a third-party platform for content optimization.

If you wantto make a mobile app stay far away from WordPress. Many of the plugins and tools will become an issue when WordPress does an update an kills everything.

Phonegap is OK but not optimal. I suggest you create a web app using the M.E.A.N. stack. If you don't have the tech talent or budget, you can submit your project to an organization like www.AlchemyLab.co and their students can develop it as part of their projects.

Yes, in fact there are various companies that do this for you for a minimal (often monthly fee)

A mobile responsive WordPress website uses a design that automatically adjusts itself to match user’s screen size. Almost all good WordPress themes are mobile responsive out of the box. You can even find great looking mobile responsive themes for WooCommerce. However, some businesses may want to offer an even nicer user experience by creating their own mobile apps. Instead of visiting your website in a browser, users will be able to launch an app from their home screen to access their favourite blog, online store, membership website, or online community. Building a mobile app typically requires programming skill, and there are plenty of development agencies that will build a mobile app for your website. However, it would cost you a lot of money. If you want to build a mobile app on a budget, then there are few WordPress plugins that can help you convert your WordPress site into a mobile app. All of them are paid solutions, but they cost way less than hiring someone to build a new app for you. Keep in mind that using these plugins still require intermediate to advanced WordPress skills. If you are an absolute beginner, then we will recommend using a mobile responsive theme and improving your WordPress speed to gain more mobile users. Here are the best plugins to convert your WordPress site into a mobile app:
1. AppPresser: AppPresser is an easy to use platform that helps you convert any WordPress site into a mobile app. They have a dedicated WordPress mobile theme that can be used to create an app for your blog, WooCommerce store, or BuddyPress website. It has a visual app customizer to easily change your app’s appearance just like you would do for a WordPress theme. You can customize colours, add custom pages, pull WordPress content, and more. AppPresser offers integrated push notifications which is easy to set up and manage.
2. MobiLoud: MobiLoud allows website owners convert their WordPress sites into mobile apps. They have two products: one for news or blog sites, and another one for websites using WooCommerce or other sophisticated plugins. It comes with push notifications, mobile advertising support, analytics, and custom branding. The app is configured, prepared, and can also be submitted to app stores for you.
3. Androapp: AndroApp is another useful plugin to convert your WordPress website into a mobile app. It comes with offline support, multiple themes, internationalization supports, unlimited push notifications, infinite scroll, and native social sharing. It does not support WooCommerce or BuddyPress. The plugin developer will help you configure and download your app which you can then submit to the play store yourself or hire plugin author to do it for you. The big downside of this plugin is that the free version allows the developer to show their own ads.
4. WPApp.Ninja: WPApp.Ninja allows you to create a mobile app or a progressive web for your WordPress website. You can purchase a single lifetime license to build your app. All app setup and configuration can be done inside your WordPress admin area. It gives you easy tools to upload your app icons, buttons, colours, translations, and more. It supports Google Analytics, built-in caching, offline content, and compatibility with all popular WordPress plugins. If you are serious about converting your WordPress site into an app, then we recommend using AppPresser or Mobiloud.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath