You cannot find these files on the App Store. Instead, you must use third-party IPA archives. Here are the most reliable sources (as of 2023):
Go to youtube.com in Safari. Then:
If your device is jailbroken (checkra1n or Phoenix for iOS 10), you can install the IPA permanently without 7-day limits.
To get a working client, users often turn to decrypted IPA files. These are versions of the app stripped of Apple's DRM, allowing them to be installed via third-party tools.
The most critical thing to understand is that the current version of YouTube (v19.x.x) requires iOS 11, iOS 12, or newer.