Shooting was done in the autumn through the winter, from August through January. The dates I could find are not sourced, but seem to be in line with other comments in this thread. Grouse shooting was officially opened August 12 and closed in November. Partridge shooting was done from September through January. Pheasant shooting was done from October through January.
Perhaps it’s pertinent to point out that there was a restriction on who could shoot (or hunt). In order to carry a gun you had to be a landowner of a certain income per year (we think it was 100 a year) and in certain parts of the country it was extremely bad manners to kill all your own birds and not leave any for your neighbour to shoot. Mr Bingley, as a new tenant, would have been well aware of this.
This begs the question of what Lydia and Mrs Bennet were thinking of, when they themselves had no right to allow Bingley or Wickham to shoot Mr Bennet’s birds – it was very presumptuous of them. Also, since Wickham was virtually penniless, it shows Lydia to be totally unaware of the realities of her situation. If Wickham kills any birds at all it will be due to the good nature of her father.