Vinegar has been used for a variety of common illnesses and symptoms for many years as a home or folk remedy. The use of folk remedies in general is fairly common. In one study of patients at a urban ambulatory clinic, 10-50% of the patients sampled used at least one folk remedy (1).
Vinegar has been commonly used to treat arthritis, diabetes, high blood pressure and head lice infestations. To my knowledge and review of the literature, no evidence has shown that the routine use of vinegar affects these conditions. Use of vinegar and other household items as folk remedies is generally safe. SOme individuals who have gastroesophageal reflux disease (GERD) or other stomach concerns, may experience a worsening of their symptoms by ingesting acidic substances such as vinegar. Other negative consequences can also occur, as in the case of a infant who treated for a lice infestation with vinegar and sustained a chemical burn (2).
My review of the literature does not mention drinking vinegar for bronchitis. In general, if someone has had respiratory symptoms such as cough or shortness of breath for more than two weeks, s/he should consider seeing a health care provider to determine if bronchitis is really the cause of the symptoms or something more serious.
References:
1. Folk remedy use in the inner city. Plotkin SR, Post R. South Med J. 1999 Aug;92(8):795-8. Accessed 3/8/11 from
http://journals.lww.com/smajournalonline/Abstract/1999/08000/Folk_Remedy_Use_in_the_Inner_City.9.aspx
2. Chemical burn caused by topical vinegar application in a newborn infant. Korkmaz A, Sahiner U, Yurdakök M. Pediatr Dermatol. 2000 Jan-Feb;17(1):34-6.