A time when criticism or other negative actions may be expected, with no mercy shown.
This derives from the period of the year when the prohibition on the blood sports of hunting and fishing is lifted. Of course, the prohibited part of the year is the closed season (originally called the close season). In the UK the beginning of the open season on grouse shooting is the 12th August - colloquially known as the Glorious Twelfth. This name was appropriated from the day of the same name (which was more commonly called simply The Twelfth, or Orangemen's Day, or the Boyne celebrations) in which the Battle of the Boyne (1690) is commemorated each year on 12th July.
The hunting season was called open season from the early 19th century, as in this example from The Edinburgh Advertiser, March 1820:
... "salmon may be got during the open season."
The figurative use of the phrase is quite a bit later. That was recorded in The New York Globe & Commercial Advertiser, July 1910:
"This surely is the open season for pitchers in the National League. Behold the clever Miner Brown being stung for eight runs in one inning."