QMK runs on any USB-capable AVR or ARM microcontroller with enough flash space - generally 32kB or more, though it will just squeeze into 16kB with most features disabled.
The following use LUFA as the USB stack:
Certain MCUs which do not have native USB will use V-USB instead:
You can also use any ARM chip with USB that ChibiOS supports. Most have plenty of flash. Known to work are: