Blood draw from the elderly or those with tiny veins
When we need to collect blood from patients we typically draw blood from peripheral veins that are superficial. These are the veins that you can see in some individuals hands and forearms. There are superficial leg veins that are visible. Drugs like nitroglycerin or other vasodilators do not have significant effects on the superficial veins that you'd be trying to withdrawal blood from in the extremities (i.e. arms and legs).  

If you wanted to really engorge someone's veins, you'd likely have to load them up with intravenous fluids. However, this would not be safe in patients needing routine blood draws; especially elderly people who might become fluid-overloaded to the point at which the develop congestive heart failure. Younger patients might temporarily have minimal increases in their circulating blood volume, but their kidneys would very quickly eliminate the excess fluid as urine.