That's an excellent question you ask regarding plaquenil. As it happens, plaquenil is not an immune suppressant, and it does not make people more susceptible to infections. Rather, it is a disease-modifying drug. In addition to ameliorating some connective tissue disease symptoms, it has actually been shown to increase longevity in lupus patients!
Best I can tell, plaquenil does not affect the veins, except perhaps in a positive way as it reduces the incidence of blood clots in people with antiphopholipid antibodies, and can be helpful in treating vasculitis. See Anti-Malarial Drugs by Johns Hopkins.
I was on plaquenil for years, but had to go off it as eventually it worsened my psoriasis. I had a very hard time going off plaquenil, because I got a lot sicker each time. You're supposed to be able to just stop it; its not like its addicting. But I simply could not, so I ended up very gradually tapering, and then increasing again when I encountered problems, and gently decreasing again, and then getting other meds to handle the gap it left. Overall it took me six months to transition off it, and to this day, I still wish I had been able to stay on it. It is pretty amazing for being so cheap!
Difficulty with veins is often caused by inadequate hydration. It can help a lot to start drinking more fluids the day before any treatment or blood draw, and to limit dehydrating fluids or medications, such as caffeine (which is in nearly all sodas), to have a little bit more salt (or not restrict it quite as much the day before), and to cut back a bit on any diuretics (like lasix) on the day of the procedure (with doctor's permission, of course).
Our blood is made primarily of water, and when we are even temporarily low on fluids, we literally have less blood, making our veins shrink and become even less accessible than usual. I find it helpful to do all of these things, and if it still looks like it may be difficult, to have a heating pad on the area for about five minutes beforehand. See Six Tips for Good Blood Tests.
Also, my personal experience is that there can be a huge difference among techs! Some of them just have the knack for difficult draws, and when I find a gem in any particular office, I get their name and try to request them the next time. The desk will tell you it doesn't matter, that they are all the same -- but in my opinion, that is absolutely not true. I do believe in the existence of Vein Whisperers, having been magically blessed with them occasionally. So part of the problem might be the tech, because some of them are lots better than others when it comes to doing veins that are not the customary, easy draw.