Perhaps I missed it, but I looked and didn't find any way to log into an Apple ID account. If you could log into an account, this would be a way to get apps onto sunsetted iOS devices. The last time I had to do this, I had to install a specific iTunes version with the App Store (which required uninstalling current iTunes); being able to do it on a website would be a hugely more convenient method.
The use case is this: You have an older iOS device; say an iPad 3, and you want to install your favorite game. However the game requires a newer version of iOS than the iPad 3 supports. So you go to the online App Store, log in with the Apple ID used on the iPad 3, and purchase the game. You can't do this on the iPad directly, because the App Store will only let you install the current version of the game, which is incompatible with your creaky iOS version.
Once the game is connected to the Apple ID, you can pull it up in the App Store on the iPad 3. It will then say it requires iOS whatever and asks if you want to install the latest compatible version instead. You say yes, and it pushes the most recent version that will run on your old device. Of course, if the older app phones home it might refuse to run because the server API has changed, or the server was shut down. But it works for some apps.
Again, though, this would require the web-based App Store to allow you to log into an Apple ID.