A lot of people have very small veins making it next to impossible to draw blood.
Would a nitroglycerin tablet (or some other vasodilator) before drawing blood help to enlarge 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.
I think the use of drugs for simple things like a blood draw should be a last resort. Having a cup of tea or two can work wonders at raising your blood volume and making the blood draw easier.
A nitroglycerin tablet would risk systemic side effects in many patients, so a topical ointment could be used to minimize these effects. Nitroglycerin has been applied topically over the venipuncture site to cause a transient vasodilation, and potentially make the blood draw easier. There are varying reports on its efficacy - Hecker et al. (1983) found nitroglycerin ointment helped venipuncture, Griffith et al. (1994) found it made no difference in vein size.