Killer Queen Black works with a wide array of 3rd party controllers.
The default first party controllers supported out of the box are:
(Note: 8BitDo controllers may experience extreme lag in local wireless play, but work well for online play)
In game you can hit L + R, then use the switch controller applet to assign the four controllers as you see fit.
source @Adam Liquid Bit#4392
Almost any fight stick that can connect via USB will work.
Here is a list of fight sticks we know for sure work:
It is possible to have one player configured in settings for keyboard, and one on gamepad.
source @mrmango#4518 on Discord
It is not currently possible for two players to play on one keyboard, as each player must be assigned a single input device.
source @CrillFactor#0592 on Discord
Killer Queen Black is confirmed to support some unexpected controllers. For example, while the original message was lost, someone said that a Logitech USB steering wheel worked correctly, and @Brice#6819 confirmed that an X-52 flight stick functioned correctly.