Is it possible to add a water storage tank to fridge that doesn't have one?
The simplest answeris to get a container with a dispenser tap and put it in the fridge. I use one of the larger Brita filtered containers, but similar things are available from several companies. This does take up some fridge space, and requires opening the door to pour yourself a glass of water. On the other hand, the added thermal mass of the water may improve the refrigerator's efficiency. 

(Yes, I've been pondering what it would take to convine a manufacturer to sell me a unit with cold water tap but no ice maker. Or to create my own.)