DIAGNOSIS The little plastic shutter has gotten misaligned in its track. A bit of debris or just forcing it can pop it out of its groove.
FIX Don't force it. Unplug the device. Use a can of compressed air to blow gently across the top of the device where the shutter slides. Then, using a dry cotton swab, try to very gently nudge the shutter back into its fully open or fully closed position to re-seat it.
IF THAT DIDN'T WORK If it's physically jammed, the internal plastic tab might be broken. The shutter is just a physical cover; the electronic camera off switch is separate. You can still disable the camera electronically in the Alexa app under Device Settings > Camera.
HEADS UP If the shutter is broken off, the underlying camera lens is very delicate. Don't try to touch or clean the actual lens with anything.