The “Robot GPS” you are referring to is a problem in robotics usually known as localization. As some pointed out you can attempt to determine the current position of your robot by applying various translations from a starting position – this is known as dead reckoning. Unfortunately, as was also mentioned, error due to slippage and inaccuracies will accumulate over time, eventually making the estimates useless.
Of course in the Vex game you might be able to get away with what is mentioned above (resetting position when certain conditions are met), but in the real world sensor data needs to be constantly integrated. Methods that use algorithms such as the Kalman Filter and the Particle Filter merge sensor data and use probability to choose a “best estimate” for the current position.
A great intro to robotics problems like localization can be found here. Implementing these methods is certainly possible, and would make for a much more interesting and real competition than the current examples of “programming skills” which consist of stationary baseball pitchers 
I’m by no means expert in the area but I do know of many resources, so please feel free to ask for clarification.