Ideally no you should not feed or let your dog drink balsamic vinegar. In small quantities it should be fine for dogs but there isn’t a lot of nutritional value in balsamic vinegar that warrants a pet owner giving this product to their dog.
It is best that if your dog happens to actually drink a bit of balsamic vinegar that you watch them closely to make sure they do not experience any health issues.
If you notice symptoms such as severe diarrhea or vomiting it is best to contact your vet right away and get your dog checked out.
However if your dog only drinks a small amount of balsamic vinegar they should be fine.
It is true that balsamic vinegar is made out of grapes and grapes in large quantities are actually toxic to dogs but it would require your dog to have a lot of balsamic vinegar in order to get sick from the grapes within the vinegar.
No, you should not feed or let your dog drink balsamic vinegar because overall it isn’t great for their health and could cause side effects to their health.
If you are concerned about your dogs health it is best to consult a vet for help.