How To Download An Ios App After Itunes 12.7

This isn't exactly what you want, but just to add this option too...

If you use the "Apple Configurator 2" app, you can still download and update apps from multiple Apple IDs - as you also could with iTunes.

Connect the device with USB
Sign in under Account:

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Choose Update in the menu:

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A more detailed description can be found here: http://krypted.com/apple-configurator/upgrade-devices-and-apps-using-apple-configurator-2/

The Apps are downloaded to:

          ~/Library/Group\ Containers/K36BKF7T3D.group.com.apple.configurator/Library/Caches/Assets/TemporaryItems/MobileApps/

Edit: iTunes 12.6.3 is no longer supported by macOS Mojave, Catalina or other later versions. The Apple support page notes this and the page is no longer being updated. Apple Configurator or other solutions must be used.

There is now another solution: Apple provides a version of iTunes (12.6.3) which supports new devices such as iPhone 8 and iPhone X which still retains the App Store.

Apple Configurator 2 works well for me, but I'm adding this answer for those who would prefer to use iTunes or can't use Apple Configurator (Windows users) Source: https://9to5mac.com/2017/10/09/apple-itunes-apps-focused-removed-stopgap/

Download: https://support.apple.com/en-us/HT208079

Edit: You can have iTunes 12.7 and iTunes 12.6.3 installed side-by-side.

Install iTunes 12.6.3 from the links above.
Copy the iTunes app and paste it to where you want, rename if necessary.
Hold Option and double click iTunes. Create a new library in a separate location. iTunes 12.6.3 and iTunes 12.7 can't use the same library because iTunes 12.7 will update the file and iTunes 12.6.3 won't be able to understand it.
Manually download the latest iTunes from Apple's website and run the installer to update your main copy of iTunes.
(Optional) Change your iTunes 12.6.3 icon to a different icon to help distinguish between the two. (Right-click, Get Info, drag new icon onto existing icon).

Important: You will need to Option-Open each time you switch versions so that you can select the correct library and don't update the older library to 12.7 format.

The Mac "Apple Configurator 2" Update button solution will work if you both have an older version on your iOS device or on your Mac, and a newer version is in the App store.

If the old version of the app is on your device, hit the "Update" button in the Configurator. After the Configurator downloads the new version, an ipa file will Temporarily appear in a sub-sub-directory of the:

          ~/Library/Group Containers/K36BKF7T3D.group.com.apple.configurator/Library/Caches/Assets/TemporaryItems/MobileApps

directory. Immediately copy the app .ipa file out of that directory to a safe place (remote storage, etc.) before doing anything else (such as quitting the Configurator app, etc.).

If an old version of the app is not on your device, but you have saved an old copy from your ~/Music/iTunes/Mobile Applications directory, or other backup, you can first use the Configurator app to install the old app on your device by using: Add button > Apps > Choose from my Mac... After installing the old version, follow the instructions above for updating to the current version of the app to get an .ipa file.

If you don't have an old version anywhere, try an Add of the same app twice, and see if you can copy the ipa file during the second Add operation if Configurator complains.