If catching is allowed, I'd take my inspiration from the
Hoberman Sphere and
related products to surround my entire robot with a cylindrical structure that starts off with lots of points facing up and down, then quickly brings both sets of points out and clamps them together to catch an incoming ball no matter what direction it came from. Reverse direction of actuation to catch balls falling from above or gather from the floor.
EDIT:
The shooter would be relegated to a secondary function, only for use in getting rid of caught dodgeballs and attacking opponents unable/unwilling to fall into that trap. Two spinning wheels on a single axle at the front of the elevator (which uses polycord belts, BTW), so that fired dodgeballs travel up and over said shooter wheel before leaving the bot in a forward direction. Not sure if I'd shorten the few catching arms directly in front of the shooter opening, or if I'd stick with closing the catcher whenever I wanted to launch an attack. Either way would be fine, because at that point it's all up to the driver.