Is your vault saved locally or in the cloud?

Your vault is not synchronised in the cloud.

Edit : since patch 1.2, it can be synchronized.

You have to enable Cloud save for your Vault in the load screen like so:

Manual backup

If you want to reinstall the game without using the cloud save, you can first make a copy of your save file - it is located in /Android/data/com.bethsoft.falloutshelter/files/VaultX.sav, either on your SD card or in the internal memory.

On iOS, Fallout Shelter save data does not survive an iPad device wipe. :( (I know this from personal experience.)

A recent iTunes backup should save the day, however.

As of patch 1.2, unofficial notes here, cloud saves are available.

Among the features is the arrival of a mysterious new stranger who can reward you with additional bottlecaps, a more difficult Survival Mode, the ability to save your Vault to the "cloud." Exclusively on the App Store for iOS, you can now get the all-new legendary character Piper from Fallout 4 and five free lunchboxes. Check out all that's new in the video above, and read below for Bethesda's official patch notes.

• New Vault-Tec network protocols are finally in place. Now you can save your Vault to the “cloud,” (no moisture condenser necessary!) and continue any Vault experiment on all of your devices.
• Some Dwellers are reporting sightings of a mysterious individual in Vaults around the Wasteland. Vault-Tec is offering a bottle cap reward for any positive identification!
• Faced all the challenges the “normal” Vault simulation can provide? Introducing Survival Mode! More danger, More glory. Make Vault-Tec proud!
• New Vault-Tec directive – Dwellers have been given authorization to scavenge equipment from dead Raiders. Gear up!
• Are you an experienced Overseer who has already undergone Vault operational training? Now you can skip the tutorial!
• In an effort to assist Overseers in running their Vaults more efficiently, Vault-Tec has added a statistics page to the Vault Dweller’s Survival Guide (VDSG).
• Support added for Russian-speaking Overseers.
• Continued general maintenance to the Vault simulation. Because when you speak, Vault-Tec listens!
• Available exclusively on the App Store, all new legendary character Piper from Fallout 4 and five FREE lunchboxes!

Bethesda forum thread also stating cloud save available.

• New Vault-Tec network protocols are finally in place. Now you can save your Vault to the "cloud," (no moisture condenser necessary!).
• Some dwellers are reporting strange, some would even say mysterious, sightings in Vaults.
• Introducing Survival Mode! More danger, More glory. Make Vault-Tec proud!
• Support added for Russian-speaking Overseers.
• Continued Maintenance to the Vault simulation. Because when you speak, Vault-Tec listens!

Overseers, to activate the "Cloud" system, you guys must mark the option on Vault List window.