There seems to be indeed not much done for this research on a definitively not patentable solution. Most online sources craving credibility for this claim mention the University of Pittsburgh. Sometimes even "a paper from". But trying to follow that lead I ended up at a page where a nurse tells just her own anecdotal story of how she herself once found great relief from sunburn by using vinegar. One of these websites was kindly transparent enough to directly mention this very weak evidence. But most try to create the false impression of a research level study conducted there to reach this conclusion.
Vinegar as a sunburn treatment is probably in use since shortly after the first Egyptian made it while suffering from being out in the sun too long. Hippocrates and Galen also mention the use of that substance for sunburn related pain and other thermal injury.
Historical evidence aside, some people try now to cash in into combining old knowledge into patentable formulations. This, however, is not backed up by rigorous scientific testing, and may very well backfire.
The effectiveness of vinegar has several important components:
for milder cases the evaporation point is low enough to provide a cooling sensation, directly counteracting the "burn"-part of the condition (higher than using water/wet towels alone).
the moisture might sooth dried out skin that will otherwise or is already cracking
- for slightly more severe cases it will also limit the growth of bacterial infections that might profit from such a situation
- a small direct reduction in pain might be difficult to be observed from the vinegar alone
This is mostly an answer from a historical viewpoint and by no means any form of medical advice. But using low concentration vinegar on mild sunburns might be quite effective. Although, if other options are available, they might be preferred, since prolomged contact with such acidic material might be an additional irritant.
Melba D. Pinnegar and Fred C. Pinnegar III: History of burn care. A survey of important changes in the topical treatment of thermal injuries, Burns (1986)12,(7).508-517 http://www.sciencedirect.com/science/article/pii/0305417986900793
Kwang Chear Lee, Kavita Joory and Naiem S. Moiemen: History of burns: The past, present and the future, Burns & Trauma 2014 2:20040169