Does autoionization occur in steam?
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.