Spares for the Nikon FE were stopped at least 15-20 years ago and what there are around now will not be for a shutter. It may be possible to get the camera serviced and the problem cleared but this may be in excess of the value of the camera itself.

My guess is the camera has NEVER been serviced and like all mechanical things that are not checked over they will eventually fail but can be rejuvenated after a good clean out and lubrication. Bear in mind your camera is at least 30 years old now. Imagine what a car would be like if it hadn't been serviced in 30 years!!!