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Depends on what you mean by a "robot" but you might look at the ZenXY machine from V1Engineering. It's an open source project meant to be used for drawing in sand but could be swapped and flipped over to use as a drawing machine. You could also Google "pen plotter projects" or something similar as I've seen a few out there that do exactly what you're asking.
As others have said, a 3D printer can do this too if you replace the extruder with a pen and configure it correctly. You're just looking for any sort of CNC controlled device for the hardware. For the software you'll need something that can generate gcode to draw what you want. Sandify is a tool that was created to work with the above mentioned ZenXY in order to create drawings and export them as gcode, but you'd need to do these ahead of time and have them stored on the device.
I'm talking a very basic robot using breadboards, icecream sticks, motors, an adruino or something like that, and making it move the arm in preconfigured motions, like circle or square, etc. Nothing fancy.
It might be a good idea to look at a pantograph for something like this, you could probably build a simple linkage out of popsicle sticks and use a couple of servos to control it. It would take some trigonometry to get it right but it's probably the easiest to build with those materials
Potentially you could even make two "arms" one with potentiometers to measure the angles at different points on the linkage and a second with servos set to reproduce those angles. So a person can trace out an object or drawing on the first one, then the second one shadows/reproduces it. It would simplify any code substantially and would probably be more impressive to the judges