Getting emulators on an iOS device is difficult due to Apple's ban on them in the App Store. However there are various ways you can make it work, some more convenient then others. My goal is to list the best and currently working methods for downloading the app, for free or paid.
The BuildStore is the best method for downloading GBA4iOS. Why? Because you pay a $11.99 USD subscription for an Apple developer certificate, which lasts a whole year. This guarantees the app will function for nearly all of the 365 days starting the day you subscribe. I say "nearly" because this method isn't perfect, and there may be a day or two when the service goes down. In my two years of using it, it's only gone down once and only lasted for about a day. All other days of the year were fine. Another benefit is you get access to more than just GBA4iOS, such as iNDS and other tweaked apps. If you plan on using the app regularly throughout the year, this service is a no-brainer, if you have less than a dollar a month to spare for such a thing. This service is ad-free and revoke-free, unlike most others below.
iEmulators provides a bunch of emulator apps for iOS. They partner with the BuildStore and are generally a trusted source. However, in the last year or so, they have added advertisements to the app in order to make money off of it. This ruins the experience in my opinion, but if you don't mind ads popping up while you're playing, and would rather not pay for the BuildStore, then this is an option. If you download from here, and any other site other than the BuildStore, the app will get revoked every week or two. That means you will have to delete it and download it again. Every time.
I haven't actually used TweakBox. I believe it allows you to download GBA4iOS, but you will still get regular revokes. I'm not sure if this version has ads or not either. If you can verify details about TweakBox, tweet at me @Circa_IXI.
Same as TweakBox, I haven't used it. If you have used it and can give me details about it, tweet at me @Circa_IXI.
Sideloading means you push the app directly from your Mac device to your iOS device by compiling the code. This is a great tutorial on how to do that. This cannot be done from a Windows or Linux computer. Must be Mac. This is an incredibly complicated process just to install an app that you have to re-sign every seven days. But since a lot of people like options, I'm adding it here.
Cydia Impactor is another form of sideloading, but can be done on a Windows computer. You need the IPA file for the app for this method. The above is a link to iOS Ninja where you can find both the IPA file and the steps to pushing it to your device.
If your app stops launching completely and you can't get it back, that means your certificate was revoked. This happens a lot if you don't pay for something like the BuildStore. Generally it happens every 7 days, but sometimes it can be longer or shorter. If this happens, you have no choice but to delete the app and reinstall a newly signed version. Before you do so, be sure to back up your save files and roms, if possible.
GBA4iOS has a built in feature that allows you to back up your saves using Dropbox, a cloud-based file sharing service. However, recent versions have been reported to not have a functioning Dropbox Sync feature, probably due to Dropbox's API being updated constantly, and nobody to update the app to make it working again. However, if you have a computer, you can easily back up all of your saves and roms via iTunes. I made a really simple video showing the steps to doing this. It shows how to back them up, but not how to restore them. To restore them, simply drag the folder back from where you took them, after you redownload a new version of the app. You can highlight all files in that window, save them all, and drag them all back. Just make sure they end up exactly in the same structure as you found them.
Video on Backing Up Saves and Roms
If your save file isn't showing up, even though you have the same ROM/game file from before, you have to match the file names with each other so they can find one another. For example if you have a Pokemon Emerald rom that isn't finding your save file after transferring them back to the app, go in via iTunes and see if they share the same name. One might be named Pokemon Emerald.gba and one might be named Pokemon - Emerald Version.sav. Doesn't matter what you choose as the correct one, just as long as they match. If one has a dash in it, add a dash to the other. If you can't rename in iTunes, you can rename in the app by tap and holding the game in the list.
The app has a built in search button, that usually doesn't work. You can change the website that the button points to in the settings. However, that button just opens a Safari window, so you can also just use Safari. There are multiple ROM websites out there, although some go down occasionally or have tons of ads. It's usually not a pleasant experience. I made my own ROM archive, and you should find nearly any game you want. Go here. I recommend just setting the button in the app to that page for easy access.
Sometimes the games download fine but don't show in the list. It's a bug that I've run into a few times. Generally if you close the app completely by going into multitasking and swiping up, then relaunching it. You can also try hard rebooting your device by holding the sleep and home buttons until it restarts. If those don't work, you may need to try reinstalling the app.
You can download custom controller skins that people have made and use in the app. I personally have made a few. You can see them here. There's also the archive made by the developer of the app that were originally included within the app itself. Take a look here. There used to be a few other good places to find skins but they disappeared. If you know of any more good archives of skins, let me know and I'll share them here.