ipod touch 4th generation compatible Virtual Reality app

Find out what company made the "cardboard device", then search the App Store. If there is an app, see what devices it supports.

However, it's highly unlikely that you'll find anything that supports a version of iOS that old.

You'll need to search the App Store for VR apps and then check to see if they will work with that old a version of iOS. But, I suspect it will be a largely futile undertaking.