How long do unrefrigerated opened canned peppers last?
Given the vinegar, these sound like pickled peppers. Pickled items are usually made to last, even when not refrigerated -- preservation was the original purpose of pickling. As it says on the Wikipedia article about pickling: 


  "Another distinguishing characteristic is a pH 4.6 or lower,[2] which is sufficient to kill most bacteria. Pickling can preserve perishable foods for months. Antimicrobial herbs and spices, such as mustard seed, garlic, cinnamon or cloves, are often added."


(Source: https://en.wikipedia.org/wiki/Pickling)

-- But, how well an item is pickled contributes greatly to its shelf-life either in or out of the fridge. 

As an example, I had store-bought kimchi (pickled cabbage) which wasn't great after a week or two opened in the fridge, but my homemade kimchi was good for months under the same conditions. 

These differences in shelf-life arise because of differences in the pH level, the ingredients used, and the conditions wherein the product was made. All of these things are controlled when buying such items commercially, which is why an expiration date is even possible. However, this is not as easily determined in the case of home-made items, because there are no strict standards on pH, ingredients, conditions, etc. 

I've made many pickled items over the course of years and my best advice is to open one of the jars, eat a pepper, and notice the taste, texture, smell, and other qualities of it. Then wait before eating another one. Draw it out and see how the taste evolves. Mark this down on the calendar and it will give you an idea of how long they may last. You can then use this when considering when to open the other jars. 

When making my own homemade pickles, this is what I do, so that I can give estimates if I give them away as gifts. Just ask the person who made them (if you can) and make sure they're all from the same batch because that can influence the overall shelf-life. If you can't do this, assuming the shelf-life is relatively similar probably shouldn't steer you too wrong. 

With homemade items, there are so many differences in ingredients and other factors that it results in a very different aging process, so a more precise answer in my experience, just isn't possible.

Just a few things to keep in mind: 


The overall shelf-life after opened may be shorter, the longer you wait from the date they were first canned. This is because the items are still aging in the sealed jar, albeit at a much slower rate than after the jar is opened. 
Take care not to put fingers, used forks, or other non-clean things into the jar when taking the peppers out. The more free of contaminants that it is, the longer the shelf-life will be after opened.