Yes, a solar attic fan can still run on cloudy days because solar panels can produce power from diffuse daylight, not only direct sun. Output is usually lower than it is on a clear day, so the fan may run at a reduced speed when the sky is heavily overcast or when the roof is shaded for part of the day.
For consistent performance in the Carolinas, the roof location matters. Carolina Skylights checks sun exposure, shading, and the attic layout before recommending placement. If your roof has limited sun, Carolina Skylights can also recommend Solatube options designed to keep ventilation steady beyond peak sun hours, including models in the ClimaSense Series that are built to run all day and continue into the evening.