The most popular camp site at Easter Island is Mihinoa. It's located just by the ocean in the town of Hanga Roa. You may bring your own tent, but also rent one.
Camping under the stars at Easter Island
According to Chilean rules, it is not allowed to camp in any of Chile's national parks. As Easter Island is declared a national park, these rules officially also apply for this island. Though, in practice it is different. As almost half of Rapa Nui is national park, this would mean that locals would basically be unable to camp in nature, which would be too much like living in a glass jar.
In practice, locals are allowed to camp in nature. If tourists are accompanied by a guide they may also camp in nature, but if a guide is not with them, they may not. If tourists are seen by park rangers camping by themselves in nature, they are removed.