If half of the rabbit population were killed then it would have an impact on the food chain but I do not believe it would last for long. The animals that feed on rabbits would have a shortage of their food supply so they would start feeding on other animals, perhaps decreasing their populations as well. In addition, the plants that rabbits eat would become overgrown. People with gardens certainly would not be sad because rabbits can do a lot of damage there.
On another note, rabbit reproduce incredibly fast and have many offspring with just one birth. It would not take a lot of time for the rabbit population to grow and return to its previous numbers.
If half the rabbits in a given population are killed off, there will be impacts both up and down the food chain.
Looking down the food chain, the things that the rabbits eat will not be eaten as much. Therefore, their population will increase. They may crowd out other plants that compete for their niche (if the rabbits didn't eat those too).
Other organisms that eat the same things as rabbits will also flourish. They will have less competition for the food.
Up the food chain, animals that depend on the rabbits for food will see their populations drop. This is because they will not have as much food to eat.