My son's tablet broke and when he gets a new one he is hoping he will be able to access all his saved worlds on his new device. Is this possible? Will the minecraft servers have his data saved if he re-installs his already purchased app?
Minecraft PE does not save information on servers, it saves it on the device's local storage. If he was playing on an Android with removable storage, the transfer is as simple as using the old card in the new device.
If the old broken device can still open Minecraft then you can transfer the world by hosting a LAN game from it, joining it with the new device and hitting
Quit and Copy World in the Pause menu.
If not, you'll have to transfer the data across using a USB cable via a computer. If the old device is broken to the point where this won't work, there's not much you can do I'm afraid.