Well...certainly, but not entirely cardboard of course., 99% would have to be an actual, functioning digital camera. You can make a pinhole camera using film, and you can make a pinhole digital camera, but a digital camera ALWAYS has a significant amount of digital electronics. The quality of the image using a pinhole for the digital camera lens will compromise image quality, and the images will be susceptible to overexposure via IR, so likely it would be best suited to nighttime imaging.Having said this, think it would be a complete waste of time, even if it is academically doable.
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Do you want to make a camera prop, or do you mean a functioning camera? The prop wouldn't be too hard to make, just cut out some card stock for each side and attach them all together. But for a functioning camera... See Re-Design's answer. He knows what he's talking about.
I think the tolerances in a digital camera are so small that it might be impossible to make one out of cardboard that would be able to keep the lens in position for proper focus. The sensors are so small that the lens to sensor is very short and a tiny movement makes a huge difference. If you can find a digital camera where the sensor and the lens mount is one piece then you could make a cardboard "box" to enclose it. If you built a camera out of a scanner, making one out of cardboard would be entirely feasible because of the distance from the sensor to the lens is great so a small movement makes little difference in the focus. Google "scanner camera" to see what has been done with them.