Is it possible to download an older version of an app if you purchased a newer one on a different phone using the same Apple ID? I have an app that I want to download but it requires iOS 11.3 or higher. That app has been available since iOS 9 and older versions are compatible with 9 and 10. 

Some apps which similarly require iOS 11+ that are in my purchased apps list enable me to download an older version of that app but the ones which i have not purchased don't. They just say that i require iOS 11.3 or higher.

Is it possible to sign into my apple ID on a different iPhone running iOS 11.3 or higher, download the app so that it appears in my purchased list and then sign out, go back to my iPhone and download the app in it's older version like the rest of the apps?
Yes, this is possible but it depends on the developer. The developer of the iOS app must enable the feature on Apple's App Store.

See How can I make allow users with older versions of iOS to download an older version of my app.

You may want to contact the developer to ask them directly.