Of course lacking sleep is possible to lead to droopy eyelid. The droopy eyelid may be the result of long term's congestion of blood capillary around your eyes, the skin would not get the nourishing and lack of oxygen, then the eyelid droopy. Frequently stay up late, emotional instability, eyes fatigue and aging will lead to the slowing down of blood circulation, the erythrocyte (red cell)'s oxygen supply will come down to lead the carbon dioxide and metabolic waste accumulation. If the blood around your eyes lack of oxygen, the blood congest and pigmentation of eyes blood will be happen. Enough sleep and relax will far beyond many nursing care for your eyes to recovery. Some eyes scream would be applied for you to care your eyes. The eyes massage, if you insist for certain period, it will improve the elasticity of your eyelid.
Yes, it is possible for you to get the droopy eyelid because of lacking sleep. The lack of sleep will make your skin become loose to some degree. Your droopy eyelid may be caused easily. You should keep the good diet and good sleep to make the droopy eyelids get recovery. You could use some essential oils to prevent the droopy eyelid.