Can the body synthesize vitamin D?

Yes. But then you have to expose yourself to the sunlight. The ultraviolet rays of the sun convert the cholesterol into cholecalciferol. The liver and kidney add hydrogen atoms at 25 and 1 position to give you 1-25- cholecalciferol. That is also called as cacitriol. This is the active form of vitamin D. Probably most of the people suffer from more or less deficiency of vitamin D. So they need to get it from nutrition supplement.

Yes, but way of exposure to the sun. Very few foods contain vitamin D so synthesis of vitamin D in the skin is the major natural source of the vitamin. Vitamins A, D, and B12 are stored in significant amounts in the human body, mainly in the liver. But after the stores are gone, there will be a deficiency condition. Well-known human vitamin deficiencies involve thiamine (beriberi), niacin (pellagra), vitamin C (scurvy), and vitamin D (rickets).