The elephants? They probably drink off the Rimfall. The turtle? Ditto.
Great picture, by the way. I'm guessing it's mostly right, because even though gravity generally pulls 'down' on the Discworld, that's only because it's flat. In reality, it might be entirely possible to walk around on A'tuin like a planetary body. This means that perhaps water will condense into clouds anywhere around the turtle - including the discworld. The water falls off, gets trapped in gravity, circulates for a while and then gets dumped somewhere else, that may, or may not, be on the disc.
On the other hand, perhaps there's some kind of magically-sourced spring? If you hit a dead-end, you can always blame stuff on magic.
It seems really strange, you know? Discussing the Science of the Discworl... oops!