There is no scientific evidence that vinegar - apple cider vinegar (ACV), specifically, since that's what is often claimed to be an effective treatment for a number of conditions - is an effective treatment for arthritis.
Some of the claimed effects of ACV:
Some proponents of apple cider vinegar claim it can cure everything
from diabetes to acid reflux, to warts, cancer, arthritis, athlete’s
foot, halitosis, high cholesterol, gout, poison oak, urinary tract
infections, and even head lice!
Not a direct answer to your question, but it's worth looking at the necessary minerals that proponents claim ACV is a good source of:
Let’s see, what else do we have? Hmm, composition… “Apple Cider
Vinegar contains cholesterol-reducing pectin and the perfect balance
of 19 minerals, including potassium, phosphorus, chlorine, sodium,
magnesium, calcium, sulfur, iron, fluorine and silicon.” (From
http://www.parrothouse.com/acv.html). This should be easy to check, as
USDA (the United States Department of Agriculture) provides an online
database of nutrient values:
http://www.nal.usda.gov/fnic/foodcomp/search/. I typed in Cider
Vinegar and found that Cider Vinegar contains the following minerals
(amounts per 100 grams):
Minerals Calcium, Ca mg 7 Iron, Fe mg 0.20 Magnesium, Mg mg 5
Phosphorus, P mg 8 Potassium, K mg 73 Sodium, Na mg 5 Zinc, Zn mg 0.04
Copper, Cu mg 0.008 Manganese, Mn mg 0.249 Selenium, Se mcg 0.1 These
minerals are present in miniscule amounts. To put this in some kind of
context, the attached PDF ‘K in food’ shows levels of potassium in
common foods. How much Cider Vinegar do you need to consume in order
to get 300mg of Potassium? 410 grams. How much baked potato would you
have to eat in order to obtian the same amount of potassium? 56 grams
(about one third of a baked potato according to the PDF).
To put it another way - the Zinc present is 0.04mg, or 40 micrograms.
This is 0.2% of the EU Recommended Daily Allowance (RDA) for Zinc. How
significant an amount is 0.2% of your RDA? Two thousandths of the
amount you require daily. Great, now I just need to find some way of
getting the other 99.8 of my RDA. (See PDF for foods rich in Zinc).
Despite the claimed benefits of ACV, there has been little research into its use as a treatment for various disorders - but in the research that has been done, no evidence has been presented that suggests it is an effective treatment for arthritis:
There doesn’t seem to be much evidence for the safety and efficacy of
Cider Vinegar on Pubmed, so I had a look around. Maybe there are some
scholarly articles not published on Pubmed that look at Cider Vinegar?
http://findarticles.com/p/articles/mi_m0GCU/is_n6_v14/ai_20152545/pg_1
seems to be sceptical regarding the much vaunted benefits of Apple
Cider Vinegar. Two quotes stand out for me:
“There is no scientific evidence that apple cider vinegar has any
medicinal properties. While the folksy anecdotes from those who claim
to have benefited from apple cider vinegar tonics may be amusing to
read, they are simply that — anecdotes.“
“The Arthritis Foundation calls vinegar a harmless, but unproven,
arthritis remedy. It points out that arthritis symptoms come and go,
and that a person using an unproven remedy may think a remedy worked
simply because they used it at a time when symptoms were going into
natural remission. Such is undoubtedly the case for many of the
“cures” connected to vinegar.”
Furthermore, consuming ACV in sufficiently large quantities may actually worsen one's health:
EDIT: Spotted this –
http://content.karger.com/produktedb/produkte.asp?typ=fulltext&file=nef80242.
Quote: “Regular ingestion of cider vinegar is becoming an increasingly
popular habit in Austria and Germany. Cider vinegar is described as a
prophylaxis and cure for almost any disease or complaint. Doses from
one teaspoon to six soupspoons per day have been recommended. A local
bookshop offered nine different specialist books on the benefits of
cider vinegar. Here we describe the case of a woman, in whom chronic
ingestion of excessive amounts of cider vinegar caused serious health
problems.” What kind of health problems? Well, the article is entitled
‘Hypokalemia, Hyperreninemia and Osteoporosis in a Patient Ingesting
Large Amounts of Cider Vinegar’. Authors: Karl Lhotta, Günther Höfle,
Rudolf Gasser, Gerd Finkenstedt. Ref: Nephron 1998;80:242-243 (DOI:
10.1159/000045180).
So, in summary: There is no scientific evidence to support apple cider vinegar as a treatment for arthritis. (The linked post goes on to address the conclusions of various studies that test its use as a treatment for other conditions such as diabetes, if you're interested. It also contains links to the PDFs mentioned in the section about minerals.)