This will not work:
(Source:**broken link removed**)
You do not want to scare the patients, particularly children.
There are vein detectors based on IR (See: **broken link removed**). There is another based on getting a negative image from scattered light (See: **broken link removed**). I suspect there are other inventions that work or don't work about as well.
As for your analysis of the problem, I would separate burn patients into a whole different category from obese patients and young children. Drug addicts are another category. Old people are easy to find veins on, but often difficult to get good access, so they are a different category too.
I recommend that you focus on the obese and young patients, with the obese patient being the first. As for needing contact to use ultrasonic methods, in theory, that should not be an absolute deterrent to the technique. Have you considered using IR?
As for hardware design, the links I gave for IR and light-based methods are are pretty intuitive. For ultrasound, I suspect you will need to do some experimentation.
John