The default way to install an iOS app was through iTunes. Having not used iTunes in several years, I'm trying to put an app on an iPad I have, and can't for the life of me find the iOS store in iTunes.
What am I missing?
You're not missing anything. Late in 2017 in iTunes 12.7, Apple removed iTunes ability to access the iOS App Store. Using iTunes 12.7, the only way to access the iOS App Store was via the iPhone itself.
Many users disliked this change, as it's harder to navigate and review apps on iOS than it had been using macOS. A month or so later, Apple retreated and offered a version of iTunes that restores iOS App Store availability. This version of iTunes is 12.6.3, and you can read more about it on this Apple page. iTunes 12.6.3 can be installed even if one has already installed 12.7. The Apple page cited above contains a link to download iTunes 12.6.3.
I am very happy to have iOS App Store access back on macOS.
You can browse iOS apps in your favorite web browser at https://itunes.apple.com/us/genre/ios/id36?mt=8. Unfortunately, you can't make purchases (or see what you've purchased, or download IPAs, or...). But at least you can view screenshots, reviews, etc on a bigger screen.
Even though they’re still keeping the build on the website, it does look a lot like a temp workaround. ‘Tech support is only for the latest versions of iTunes’ also is a bit of a red flag.
Since Apple is clearly dropping this feature, I thought I’d get into the thread and tell you about our solution. We have app pretty great app management features in our iOS management software iMazing, including download of IPAs. This functionality is 100% free, and followed the same paths iTunes did, so it’s also 100% legit (no download if you haven’t bought the app already). More info about it in our blog: https://imazing.com/blog/imazing-2-5-download-apps