The mobile version of Triple Town can be played without an internet connection; however, certain actions (purchasing coins, restoring non-consumable purchases, etc.) do require a connection to the iTunes / Amazon / Google Play servers.
It is sometimes possible that Triple Town isn't able to connect to those servers even if you have an internet connection and even if other apps on your device are able to do so. To determine whether you're in this situation, please follow these screenshots:
If you find that you're not connected to the app market, this page may help you to fix the problem.