I'm not sure if the FAWN units are custom boards or not -- but from one of the photos, it seems they use AMD Geode SC1100 CPUs; beyond that I can't really find out much about them.
Those motherboard don't so much have "onboard power supplies" -- as they accept straight DC (possibly 12V?) from a wall power adapter with a barrel connector.
Most power supplies convert wall AC power to DC power used inside your computer. Typical ATX computers use a larger ATX power supply that supplies hundreds of watts, are larger and generate more heat. These FAWN units only sip power, thus they can get away with a small wallwart and don't need a typical ATX power supply.
ZOTAC makes an nVidia ION board that similarly doesn't require an ATX-style power supply, but use a power brick that you usually see with laptops.
(see the barrel connector for power on the far-right edge next to the audio, and under the Wi-Fi antenna connector)