Neat idea!
What you really need is a battle bot minus the weapons of mass destruction and crazy armor. Doubt you could get it for $100 though.
R/C cars with tank steering would require two ESCs and a tank capable radio. Or some other mechanical means of accomplishing the same thing.
And, to keep cost and speed down, I'd just use a single really high turn brushed motor (crawler) and gear as needed with whatever voltage you want to run.