No there's not a lot of sugar in balsamic vinegar.
Balsamic Vinegar is pretty low in sugar content and also low in fat so it's okay to use Balsamic vinegar in foods when you're on a diet or when you need to watch your fat intake as well as sugar intake.
Balsamic Vinegar contains many health benefits as well due to the antioxidant called polyphenols which fights cell damage and boost your bodies immune system.
The balsamic vinegar can help reduce your cholesterol levels as well as protect against heart disease, cancer, and other inflammatory conditions.
So by using and consuming some balsamic vinegar you'll improve your health.