The answer by LarissaGodzilla is very good. But I would like to add one thing that you can do that actually increases your vascularity, independent of genetics (which nevertheless influences the extent of this effect).
Doing interval training, such as HIIT, increases vasogenic factors into the blood, which subsequently causes blood vessels to proliferate. (ncbi.nlm.nih.gov/pubmed/8790589, ncbi.nlm.nih.gov/pubmed/19258658) When doing long distance running, this effect is somewhat lower, and occurs more centrally in the muscle (in order to supply more blood and oxygen for energy). However, in intensive interval training you activate the muscles forcefully, which presses the blood from the muscles to the superficial veins. That increases the pressure inside them, which acts as a stimulant for vasodilation and vasogenesis in itself. But it also increases the delivery of vasogenic factors specifically to those veins, which leads to further vasogenesis.
I would also like to add an explanation to LarissaGodzillas argument on temperature. It is indeed the temperature that causes the vasodilation. That is why it is bad for people with varicose veins to shower with hot water. When the temperature is increased, the body responds by vasodilation. This moves the blood superficially, so it can emit more heat and as such prevent overheating.