[NoodleKnight]

You'll have a hard time. For starters, the mechanical ABS module is of almost no use to your robot, it was designed specifically for that car and well, no competition legal robot will come close.
Same goes for the electronic module, it's not worth the time and effort to make one work with your robot. It too was specifically designed for a specific car, for that specific car's brake system. It's unique to that car and it's handling characteristics. Moreover the ABS module was designed to control the ABS solenoids to pulse the brakes. I wouldn't advise against putting brakes on your robot. Beyond the extra complexities and extra weight, you can use your electric motor to achieve the same effect.

Like mentioned above, I'd stick to trying to create something in programming. Most modern ABS systems sample individual wheel speed and use that data to determine whether or not a wheel is slipping. Most wheel speed sensors are just hall effect sensors that 'watch' a toothed wheel on the axle or wheel hub.

How you determine wheel lock up can vary, there's probably manufacturers trade secrets in regards to this. Two ways that come to mind are comparing individual wheel speeds, or looking at how fast the wheel speeds have changed. From there you can start making things more interesting and see if they yield nice results.
I've given the idea of an ABS-like system myself and I think that just controlling the increase and decrease of drive throttle is probably as far as I'd go. Beyond that it'll start becoming intrustive for the driver and will only partically fix your slip-n-slide problems.

It's hard to compare cars to robots since even though they have wheels and a motor powering them, they are used differently and move differently. But for what it's worth, in low friction environments the best ABS system won't stop your car from sliding sideways, or in any direction once it gets going. First hand experience, lol.