Anyone know if and where I can buy a water distiller that can dispense cold water?

Akan-

I'm not familiar with such a unit. Your machine would need to contain both a boiler to do the distilling, and a refrigerator to do the cooling.

I have an old Sears one-gallon distiller. It takes around 6 hours to do one gallon.

I see a similar unit on Amazon, "Mophorn Countertop Pure Water Distillation Purifier with Handle, 4L, 750W, White", for $69.99. There are fancier models costing two or three hundred dollars, but they work the same.

Fred

I suppose it could be done with some kind of heat pump but I doubt the ROI would make much sense.

I don't think this is what the OP wants, even though it seems to be just what he wants.

Temp.Range: 0?~35? Cooling Capacity: 500W@25? Pump Capacity: 3.5L/Min@0.5bar PID temperature control feature allows better temperature control, greater energy efficiency, and longer compressor life time CFC-free refrigerant R134A cryogen is compatible with international environmental standards.

That chills the water but it does not dispense it but only re-circulates it. Nor does it distil water but is used on a distiller.

Collect your distilled water in a container, put it in the refrigerator, wait a while, done!!

you also have to have

Circulation is very important.

Yes, I mentioned that, minus one word.