I'm not exactly sure if it's specifically a deer hunting game, but it is at least a hunting game. I played it in the early 2000s on either Windows 98 or Windows XP. The graphics were in 2D. There was a shop you could buy stuff from, and the hunter could set up camp. I faintly remember there being choices for buying sleeping bags, etc. There were quicktime events where the game would play a video file with deer running, and you would have to drag a crosshair around to shoot them.
The 2D graphics were a top-down view (?) and there was ambient music of birds chirping (?).
Once the animal was dead, the hunter would see a dead animal with blood splattered on it in the 2D world. That's all I can think of right now, thanks for reading.
I searched up "Deer Hunter" on Gamefaqs, but the games are so old and obscure there aren't many images out.