Depending on what's causing the problem sometimes vinegar works and sometimes it doesn't. The reason it works is that it changes the body's pH, and makes it more acid, some dog needs to have it changed the other way, and make it more alkaline, and Tums works for these dogs. You don't want to do both, or you won't be doing anything. Sometimes just doing this for a short while returns the dog's system to the proper pH, and then you can skip them treatment. Distilled water without anything added works for many dogs, as does various other treatments. This link has great information on tearing and how to get rid of stains.
Tear Staining: The Causes & Cures, however it says "white cider vinegar, lol, and I thought cider was the yellow kind. It probably doesn't make a difference, both kinds of vinegar are safe to eat, and both are acid.
Tear Staining: The Causes & Cures