There are many possible designs, let them experiment (oh the days of 2x20 beams flying across the classroom…)
The easiest that comes to mind is a rotating element (short beam) pulling a, let’s say horizontal, linear-motion element. The end of the rotating part, in the arch segment of 60-120deg, is moving almost in a line, but beyond 140deg, it goes away vertically, releasing the horizontal, rubber-band-loaded element.
Another option, well published, is the the Hexbug hook shot ball machine, see https://www.hexbug.com/vex/hexbug-vex-robotics-hook-shot.html.
I bought that for my kids and they did one ball shooting design based on that 3x120deg rotating piece. No need to buy (though I think it’s great value for the money), you can let them inspect the downloadable build guide.
Another, long abandoned design from my team was a conveyor belt (vex chain treads with widely-spaced intake flaps), where the flaps were pulling the rubber-loaded element. As the element hit the hard stop, the flap bent and released it. The beauty (and a great deal of complexity) of that design was that it was synchronous - the belt was an elevator, one flap passing = one ball seated in the firing position, but also one flap passing = one shot. They have had troubles with proper timing of the release and also with the rigidity of the overall construction.
I have also published one concept for quick energy release at http://www.vexiqforum.com/forum/main-category/vex-iq-robot-showcase/11039-catapult-bot-on-wheels
It’s a concept, not a ready-made design, so I think it should be a fair game for a competition team, they’d need their creativity to turn it into something useful.
To get most of that for the students, don’t forget to sprinkle in a bit of physics - there are online canon shooting simulators predicting the ball trajectory based on angle and initial velocity, and the theory they’ll gain will match the reality surprisingly well in this small scale of 3" balls and 20-30" of trajectory…