Vapour pressure is normally defined as an equilibrium phenomenon. In statistical mechanics terms it is the point where an equal number of molecules are leaving the liquid and entering the vapour and leaving the vapour and entering the liquid.
This means that, however small the exposed surface, enough molecules have gone into the vapour phase to create the equilibrium. But most systems are not observed at equilibrium. The kinetics of the process leading to equilibrium clearly depend on the surface area of the liquid as there is far more opportunity for molecules to escape into the vapour phase if the exposed area is larger. But, if you are prepared to wait for the equilibrium to be established, the surface area doesn't matter.
Hence the slight intuitive confusion: what we mostly observe depends on surface area because what we observe is mostly systems that have not yet reached equilibrium so surface area matters. But on a strict equilibrium definition surface area doesn't matter but you might have to wait a long time to see the eventual equilibrium position.