I built a fairly simple application in Unity that uses a webcam and opencv to overlay graphics in realtime to a user's face. I am looking for a low cost microcontroller/computer to run the installation and output it to a monitor. I was originally thinking Pi but Unity will only build for x86 architecture. I saw the Udoo x86 but it is ~$200. The Intel Quark seems to be x86, but I'm not sure if it could handle the graphics. Does anyone know if it's possible on this board? Or what specifications I should look for to determine this?
Most of the Quark parts are effectively small microcontrollers. For example, the Quark D2000 that you linked to runs at 32 MHz, and its working memory is limited to 8 KB of internal SRAM. This is completely inadequate for any kind of image processing, even if it had any way of driving a display output (which it does not).
The Quark X series (e.g, Quark X1000) runs at a more useful speed (400 MHz) and supports external DDR3 memory. However, this series was released in 2014 and suffers from some serious CPU errata which make most standard Linux software unusable. If cost is a concern, I'd recommend that you look into alternative software that will run on an ARM SoC.
If you run a common CPU 2/4 at 1.6Ghz single frames may stream after 20-80 miliseconds each with a shape detection and overlay.. Quark is quarter speed apparently... it may be little slower depending your thread. But interestingly an overlay may take just 2 miliseconds thanks to internal block memory copies. Thus an overlay only application may run full frame on Quark depending your streaming and color decoding strategies and overcoming compilation issues.