According to a report by Eurogamer, the Nintendo Switch will be able to play GameCube games through its Virtual Console service. Anonymous sources have confirmed Virtual Console support for three titles: Super Mario Sunshine, Luigi's Mansion, and Super Smash Bros. Melee. Animal Crossing on GameCube is also being tested apparently.
The Virtual Console program has brought titles from the NES, Super NES, Master System, Genesis, TurboGrafx-16, Game Boy, Game Boy Color, Game Boy Advance, Game Gear, and MSX to the Wii, Wii U, and Nintendo 3DS. This is the first time GameCube support has been offered. Nintendo is reportedly looking into allowing previous Virtual Console purchases to be accessed on Switch via a small fee, similar to NES VC title support on Wii U.
Nintendo European Research and Development, the group behind the NES Classic Edition, is behind the Switch Virtual Console. Nintendo is also working on Switch support of the Wii U's GameCube controller adapter, but the final decision on that front hasn't been made.