That's a fun question. I think the answer is Yes. You need to figure out how to get the gas from the reaction into a balloon - maybe you could put the baking soda and vinegar into a small soda bottle and try to get the balloon on the top fast. But you might not get enough pressure to stretch out the balloon, depending on how much baking soda and vinegar you used and how tightly the balloon was attached to the soda bottle.

Baking soda and acid (any acid, in this case acetic acid) will react chemically to produce carbon dioxide and water. Whether or not it causes a balloon to explode depends on the amount of gas being put in.