Is it possible to make a standalone game using the GECK?

Or does it have to be connected to the Fallout/New Vegas wastelands?

The idea is to have my own vault and rather then having to wander the wastelands to find my vault I would just like to start there.

It is entirely possible to make a new game, if by that you mean a whole new world with quests and the such and is not connected to the wasteland at all. It would just take time and a lot of patience. I would suggest you find out more about those mods that are alternative start mods, as you can use that to be able to start where you want in your "New Game" with the new tutorial.

To be highly technical, "standalone" means "runs as an EXE on its own"... the GECK cannot do that. But it can make an entirely new master data file that will use the FO:NV launcher and engine but draw data from your master file only. Your maps, your items your NPCs your scripts, nothing from FO:NV used except the texture/sound/model files and engine itself.