A peristaltic pump mechanism driven by a stepper would be perfect. If printed, the mechanism should probably be in nylon so it doesn't go out of tolerance after awhile. (2nd choice PLA, 3rd choice ABS, based on the hardness of the plastic.) Make sure you buy tubing that's marked Food Grade.
To keep the drinks nice and cold, you can either use Peltiers, or phase-change cooling (like air conditioning). There are refrigerated PC cases that you could use for a chassis. The compressor is in the bottom of the case, and of course they have provisions to mount a 5 & 12V PSU, which would be convenient.
Peltier: Much easier to design a customized heat rejection system. Phase change-cooled cases have relatively fixed assumptions (your CPU is here, your graphics cards are there) that you'd have to work around. With Peltiers, you can site the "cold sides" of the system anywhere you want. You will need to use antifreeze, surfactants, biocidal agents, and possibly something else I'm forgetting, to make sure the tubes and water blocks don't turn into a science project. One idea might be to make a casque to hold a bottle of "feedstock," with copper tubing to run chilled water through wound around the inside in a helix, perhaps cemented in some copper compound (not RTV, but something designed to be conductive) to increase the rate of heat exchange. (You could just sit a Peltier at the bottom of the casque, but it would cool the bottom first and the top last, encouraging a temperature gradient.) The internal beverage containers would be removable. You'd clean them daily, and when you put them back into the casques, they'd fit snugly with the wall. The fluid running through the tubing would be routed to a waterblock (USE COPPER, NOT STEEL OR ALUMINUM). The waterblock would be sandwiched to the cold side of one or two high-wattage Peltier junctions, which should be potted with rubber cement in a low-humidity environment to prevent condensation from wrecking the junctions. On the hot side of the Peltiers, you'd have a large COPPER heatsink with a fan. If you use something other than copper, you'll only make it harder for the system to do its job.
Phase change: Harder to design, as you'd have to work with fixed assumptions, i.e. the CPU is
here and the video cards are
there. However, Puron, or whatever they replaced R42 with, is WAAAAY better at carrying heat than water, even if you add antifreeze and surfactants to it. These systems are professionally charged and sealed, so there's less risk that the coolant would leak. The tubing-in-a-casque system above might work if you use a Peltier to pump heat from the water into the cold side (evaporator) of the refrigerant system. In fact, it will take heat out of that system FAR more quickly than the air will take heat out of a heatsink/fan combination. There's also the question of whether the phase-change system lets you adjust the target temperature, and by how much.
For pumping, displacing the fluid with air, as suggested above, sounds like a good idea to me. You'd need to make sure to use something definitely capable of pumping air with a positive displacement (like a Roots supercharger) as it's necessary to precisely control the amount of fluid that's displaced. Air is harder to pump than liquid because it has so much less density. It might also be necessary to use barometric and heat sensors, so you know you're always pumping the same amount regardless of weather, altitude, and temperature.
The other (easier) way to do it would be to drop a tube into the bottom of the container and use a peristaltic pump on a stepper. You might be able to clean it by filling the internal tank with hot soapy water and telling the machine to drain the whole thing. You'd have to clean some parts of the machine daily anyhow, so this probably wouldn't be a big deal.
If you use water cooling in any fashion, wrap the nylon tubing with pipe wrap so you don't waste half the energy cooling the inside of the machine.
For filling cups of different diameters, you might be able to use an IR probe (like dc42's) or ultrasonic rangefinder (with sufficiently small minimum sensing distance) to ensure a consistent fill. With that, you could do the air-displacement system without having to worry about temp/barometric pressure.
I hope you do this, and show us the results!
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