I'm looking for a way to put a small lamp inside a piece of furniture through which I do not want to drill any holes or run wires along. Are there any premade induction power systems that can get a few watts across less than half an inch separation? I'm flexible about the voltage and whether it ends up as AC or DC since I can just choose a lamp appropriate for it.
It is possible, but the energy losses are great, and the insulator (your desk) will heat up, probably causing heat damage. Even then, the receiving coil wouldn't be integrated into the base of the lamp.
In short, there's no way this wouldn't be even more obtrusive than a cord hanging off the side of your desk. If the desk is against a wall, probably better to put in a desk height electrical outlet.
The reason induction charging works for cell phones is due to tiny air gaps, low power requirements, and generally good heat dissipation compared to energy losses. Only a tiny fraction of the used power actually makes it to the cell phone. But the phone is storing that, so eventually (over the course of an evening) the phone is charged. By the way, most phones lack receiving coils, so you still have to plug in a receiving coil and wire into the phone charging jack.
Power that is not directed tends to dissipate according to the inverse square law. Even if it didn't, the losses in transforming the power to wireless and back are quite large. Some of the wireless community is quick to point out the wire transmission losses, but that's nothing compared to wireless.
With a lamp, you'll need a lot more power, and continuous power, so the issues with induction fed systems will become a lot more apparent. I would recommend a battery powered LED desk lamp (with a set of rechargeable batteries). The LEDs will extend the time between recharges.
There are inductive cell phone chargers, however these will probably give at most a watt of power only. If you need more power than a cell phone charger can give you could consider cannibalizing an old induction stove or electric tea pot.
In general, the power you get out is necessarily AC due to the physics of induction and will probably be inefficient.