When was the color camera invented?

1861: Scottish physicist James Clerk-Maxwell demonstrates a color photography system involving three black and white photographs, each taken through a red, green, or blue filter. The photos were turned into lantern slides and projected in registration with the same color filters. This is the "color separation" method.

1868: Ducas de Hauron publishes a book proposing a variety of methods for color photography.
1906: Availability of panchromatic black and white film and therefore high quality color separation color photography. J.P. Morgan finances Edward Curtis to document the traditional culture of the North American Indian.

1907: First commercial color film, the Autochrome plates, manufactured by Lumiere brothers in France

1936: Development of Kodachrome, the first color multi-layered color film; development of Exakta, pioneering 35mm single-lens reflex (SLR) camera

Timeline information from:
http://photo.net/history/timeline

This is a tricky question because you can say that cameras have been around a long time enabling Black & White prints, and they were just waiting for colour plates to be invented that could fit inside the camera, thus producing a colour image and thus turning it officially into a "colour camera" able to produce proper single colour images.

Earlier attempts involved taking three black & white plate images; each through a separate colour filter. A colour image was produced by projecting these three images through the same colour filters used to take them. So you can see why I don't consider this a true colour photograph.

Therefore, I would say the colour camera was invented when colour plates were developed.

This distinction goes to the Lumiere brothers from France whom in 1907 developed the first commercial color film called Autochrome plates.