[RunawayEngineer]

Is there another document that describes the performance requirements of the machine? Top speed, range of shot, sweeping angle of shooter, trajectory angle(s) of shooter, maneuverability, etc. I don't see how you can design without at least a couple of these. "Driving places the fastest" doesn't narrow things down.
What is the reason that you are barring using chains/belts outside the drivetrain? I am assuming that it is to make the machine more reliable - and therefore to get practice in designing a competition machine that is reliable. Personally, I find it counter-productive to ban solutions to problems because examining different solutions is a key part of the engineering process. I know that you have to draw lines somewhere - but belts/chains are not only extremely useful in FIRST, but in industry, as well. The off-season is the perfect time to test out your assumptions.
That being said - I think that it is a great idea to use a fun project to teach/practice DFMA.