I suspect more than just the relays will need to be switched, but would need higher-resolution pictures from more angles or a listing of the numbers on the chips. There are several larger surface-mount components which are likely MOSFETs, as well as a few ICs that may make up the PWM control function that gives you the variable speed.
What's unlikely is that there is any regulation for any of it (to drop the voltage down). So at the very least when you put 18 or 24V, you'll fry the relay coils in short order, let alone the semiconductor circuitry (if it's not rated for higher voltages). It may be possible to make edits to allow it to use two supplies -- one from a regulator set to 12V, and the other for the actual motor voltage. It almost certainly wouldn't be a trivial modification, and without better pictures it's impossible to say whether it's even feasible.
FWIW, the image is just good enough that I can make out the PWM controller, which appears to be a TI DRV8704, which is good up to 52V, so it won't be the limiter:http://www.ti.com/product/DRV8704/support
I can't make out markings on any of the other chips.
F-150 - 24-36V with homebrew 100A variable-speed motor controller, 4x 775 motors (no, it's NOT all-wheel-drive--look HERE
for more details!)
Pink Rocket Princess Mustang - 15mph, 36V, 4x 775 motors, homebrew PWM controller rev 2....
Hurricane - 24V w/50A step-down controller, 775 motors