Does autoionization occur in steam? It is well known that H2O spontaneously autoionizes to H+ and OH- when it is in liquid phase. I was wondering whether it does the same in gas phase, either to the same extent as in liquid phase or at all.
Yes, water is ionized to some extent in all phases. One study even extends to 1,000 K!

A concise discussion of the change in ionization with temperature at Chemguide also explains the variation of pH of neutral water with temperature. [The concept of pH 7 being neutral is based on Kw (in mol2 dm-6) of ~10-14 at ~300 K.]

In general, the higher the temperature, the more a compound is ionized.