I just started playing TOMB RAIDER (2013) for twenty minutes or so. I made it out of the cave and am running around in a beautiful forest. I have found the bow.
Lara claims to be hungry and the game suggests to kill a deer. I ran around a bit and found a bush of berries which disappeared after pressing E, so I presume Lara ate the berries. But there still is no way out of the forest and the task to feed Lara is still unsolved.
Is there a way to continue playing the game, without murdering a deer?
No, this is part of the game. Also, hunting animals as a valuable way in the game to earn materials to upgrade your equipment. Especially on higher difficulties, there is almost no way to ignore animals. You need these materials. So the game probably isn't for you.
Just in case you are a vegetarian or an animal welfare activist - Learn to seperate a game from real life. You don't kill a real animal, and you probably never have to. Mixing up real life and gaming is not a good idea.
There is no way to pass killing the deer. Though it might be sad, that's the story of Lara and the way of the game. If you have an issue with killing a deer than I'm not sure you should play this because it contains it's fair share of blood and gore. I understand why you might not want to but it's part of the game and remember it's fake. Hope you end up liking the game though, I found it quite enjoyable.