What! How did you three post before I did? There were no posts when I decided to reply! Oh well. Here's my post anyhow: [:D]
It'd be easy. All you need is an adapter for the extension cord to plug two things into it.
Get a cord, and a light bulb socket (you can use the small ones like they use on chandeliers and things like that, Home Depot (not Lowe's)).
Run the wire up to the ceiling of the safe and wire both ends into the socket, affix the socket to the ceiling.
Now, at some convenient position along the cord, cut one of the wires (not both!) and put a refrigerator-type switch in between both ends of the cord you cut, and affix that switch somewhere so the door pushes it in and turns off the light when it closes.
Then put in a 15 watt bulb or something in it, and plug in the cord.
That should work, but it's just a generic set of instructions. Sorry I can't be more specific since electronics really isn't my bag.
Make sure everything you use is rated for 120vAC currents (for example, don't use a 12v switch).
Or, you could use one of those 12v adapters and wire up a 12v light system and plug the adapter into the extension cord. Home Depot has various kinds of 12v lighting kits that might work. You'll probably have to wire in the pressure switch yourself, but that's easy.
I would double check to see if it matters which "side" of the current the switch is on, I don't really know if AC stuff can work in either direction, of it's one-way only like DC (for example, putting batteries backwards into a flashlight). Check on that. But the rest should be basically (generically) as I described above.
Maybe someone with more knowledge of electonics will post and fill in the blanks.