Which vessels allow diffusion of oxygen and nutrients across their cell layers?

The correct answer is "capillaries".

Capillaries are small blood vessels that is composed of a single layer of endothelial cells. The thinness of the capillaries as well as the thin single layer of endothelium will allow the diffusion of oxygen and nutrients across their cell layers. Some capillaries are fenestrated (i. e. kidneys) or sinusoidal (i. e. liver) which also contribute to the diffusion of oxygen and nutrients.