Older apps which are no longer supported that have an online component will eventually fail at some point due to changes in the Internet and any proprietary servers that may be required to utilize the app.
If you reinstall the OS, then it may be impossible to reinstall some third party apps if they require online product activation because many of those developers turn off those activation servers. It is one of the dangers of using proprietary apps which require product activation licensing servers. As long as you don't need to reinstall the OS or the app, then these apps will usually continue to work as long as they don't require any online component for use.
Some developers may continue to provide older versions of their apps for download & installation, but this is becoming rare these days. If you want to continue using older systems, then you should retain the installer or archive the app so you can put the app back onto your system (as long as it doesn't require online activation).
Many app stores will allow you to redownload apps you have previously purchased, but many companies seem to change the rules later on so if you can archive an app, then you should do so if you wish to be able to access that particular version of an app. However, by default, these app stores will only show the most recent version which may not be currently compatible with an older OS or computer hardware model, but it does not mean the older version is not available if previously "purchased".
Apple and most developers are only looking to the future and current OS versions so keep this in mind. As others have mentioned, this occurs for every OS out there and most apps.