London Underground: Do Waterloo & City Line Trains Have Windscreen Wipers?

Yes they do (see pic linked to below).
Not sure why though as the line is entirely underground, depot and all. However, I imagine that the reason is because they are identical to the 1992TS used on the Central Line. As this runs for significant distances over ground there is a need for them to have wipers, so the W&C stock had them too.
Then again the tunnels do tend to leak a fair bit, which is one possible reason why the W&C Line became known as “The Drain”. Therefore, because the tunnels are unlikely to be completely dry, there is still a need for windscreen wipers to clear water even in a line running completely underground.