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:
There is limited support for one of Atmel's ATSAM microcontrollers, that being the ATSAMD51J18A used by the Massdrop keyboards.
ChibiOS-Contrib has support for the GigaDevice GD32VF103 series microcontrollers and provides configurations for the SiPeed Longan Nano development board that uses this microcontroller. It is largely pin and feature compatible with STM32F103 and STM32F303 microcontrollers.