Here's what we know:
* The Games are held in a dome-like arena: This means the weather is manipulated, and the Games could occur at any time of year.
* The Games are held annually: This suggests they don't always occur in the same season.
* Katniss hunts in the woods: This suggests the setting is somewhere that experiences at least some degree of seasonal change.
Ultimately, the "season" of the Hunger Games is left open to interpretation. It's likely the authors chose not to emphasize a specific season to focus on the broader themes of oppression, rebellion, and survival.