Does app store work on ios 6.1.6?

I guess you are talking about an iPod Touch, but which version (perhaps 4th generation)?

iOS 6 is now 9 years old.
It seems very unlikely that any current apps will run on such an old version of iOS.

The fact that you can't even launch the App Store seems to support this view.

You should double-check if your device will run a more recent version of iOS.
If not, then I think it's game over.

It still works go to a game that you want with any app go to your current iPhone or iPad and install the game you want or app and then it will show up in your library or in the automatically install if you don’t have a current iPhone, you have to go on computer windows or Mac go to iTunes or download iTunes and then go to your apps if they’re in there if you don’t have an Apple ID account create it, or use your other devices to sign in I hope this is useful

Hi there,

You might have better luck asking this question over in Apple Support Communities run by Apple Support.