How do moving iron meters work on AC current?
The moving iron part is attracted by an electromagnet and the current through the electromagnet determines the physical pull/force on that iron. The polarity of the AC current has no importance; just like a piece of iron is attracted to a magnet irrespective of whether the north pole is closest or the south pole.
Both half cycles of the AC current will attract the magnet.

Won't the vane keeps oscillating when connected to AC?

If the moving iron part were in fact magnetized, then polarity would make a big difference but, it isn't magnetized hence it is attracted on both half cycles of the AC waveform.

What does the moving iron voltmeter measure in a circuit, average voltage or RMS voltage?

What is the reason that moving iron instrument can measure both AC and DC but moving coil instrument can't?