When a mosquito bites it gets the blood from capillaries, which deliver blood to the skin. Since the skin is living tissue that means capillaries are delivering blood to it everywhere. That would suggest to me that they just use their "needle" and hit the blood they're after. I'm don't, however, have medical training; so this answer is only a somewhat educated guess.
Capillaries are must smaller than arteries or veins; so mosquitoes don't have the same challenge as, say, a lab person who has to find a vein to take a blood sample.