This is even better if you switch the outlet on and off when the drawer is open/closed, that way no one is left wondering if they left a curling iron/etc turned on...

@WEDDELLKW excellent additional detail!

I think this is a really creative idea. However, it may have trouble passing an electrical inspection because I think it violates the National Electric Code. One part off the top of my head is 314.23 (A) Surface Mounting. ...shall be rigidly and securely fastened in place.
Because the drawer will be opened and closed often, it is very likely that the building wire (Romex) will be flexed back and forth which could cut through the insulation and cause a short circuit.

I don't know about the US, but here in the UK there are a bunch of British and International standards that need to be complied with for electrical wiring:

http://en.wikipedia.org/wiki/Electrical_wiring_in_the_United_Kingdom

I'd be surprised if this installation was allowed. That looks like a wall-mounted socket, but what's going on behind? Is there an enclosure that would stop stray bits of metal coming into contact with live wires? Of course if you did it competently there would be, but actually, what passes as competent in an unusual scenario like this? And this looks like it's in a kitchen where you might spill water - into a piece of furniture that might not be water-tight? Can you be certain that water wouldn't drain into this drawer, pool, and be touched by someone who sticks their hand in? And the concealed cables at the back that are constantly moved as the drawer is opened and closed - you would never know if they had frayed until something shorts or you have an accident.

Its been too long since i went to trade school (class of '90), but i'd be suspicious of this being allowed by the code. Good idea, but household wiring isn't very flexible, not to mention that plastic electric boxes don't clamp tightly to the cable. Still, you can mostly dance around this by essentially making this an extension cord, put an outlet under the vanity, and plug this into it, the outlet will be GFCI proctected if its hooked into the bathroom's wiring properly, which'll all but eliminate the shock hazard, and even a heavy duty cord is fairly flexible. Then its just a matter of coming up with a good choice for an electrical box.

Love the fact that this post follows the one of all the extension cord warnings! Cool but probably not a good idea over time!

This seems like an immediate red flag to me, too. I highly doubt that you can have an electrical box on a non-fixed surface.

But, you could put an outlet inside the base in a door/drawer cabinet, then put a grommet in the back or bottom of your drawer and run a cord through that, kind of like a cable grommet in a desk top for your computer.

Love it but I have to agree with everyone else. I think I would get around the issue by cutting out part of the drawer in the back and mounting the plug under the top of the surface. Still not to code but it would be far safer if you're really set on the idea