Moon exploration is currently done trough anomaly reduction:

In a way it has been.

The effect of this is that sometimes the +1 pop moon turns into an additional +1 industry per pop.

So this is moon exploration but maybe more is coming.

Vodyani seem to be unable to explore moons/reduce anomalies. Is this intentional, or a bug?

voydani can't access the planet modification panel (anomaly reduction/terraforming/planet improvements) this is intentional i beliive as they are not living on the planets but on the arks)

Even if they live on the Arcs they should still be able to reduce anomalies as they are affected by them.

Or they shouldn't be affected by them at all (both positive and negative effects) but that wouldn't make much sense for the ones like Rich Soil and so on.