Oh yes. It is very possible and easy to do. We make our own wine vinegars, both red and white, from left over lees. Add some herbs, and you have an expensive herbal vinegar. Here is a great link for making the mother for the vinegar. Making a Vinegar Mother. But first – What is a vinegar mother? Wikipedia explains it this way,

[A vinegar mother] is a substance composed of a form of cellulose and acetic acid bacteria that develops on fermenting alcoholic liquids, which turns alcohol into acetic acid with the help of oxygen from the air. It is added to wine, cider, or other alcoholic liquids to produce vinegar.

And from Mossgrownstone, we learn that

The resulting vinegar tastes great- use as you would use store brought apple cider vinegar. It takes less than 5 minutes to make. I make about 2-3 batches a year- making about a half gallon at a time. Other than the initial cost of the bottle of Braggs vinegar with the mother my only cost is the apple juice. I cook and clean with the vinegar but I do not do my canning with it. In order to can with homemade vinegar it is important to get a hydrometer (anywhere with homebrew supplies should carry one) in order to insure the vinegar is acidic enough.