In a vinegar and baking soda balloon experiment, which gas law is being demonstrated?

That would be , which states that when pressure and temperature are kept constant, the volume of a gas is directly proportional to the number of moles of that gas present.
In other words, the more gas you evolve, the bigger the volume of the balloon will be.
The classic experiment features a balloon placed on top of a bottle in which the reaction takes place. The gas evolved from baking soda and vinegar is carbon dioxide, or ##CO_2##.
Assuming your reaction takes place at constant pressure and temperature (in the chemistry lab or at home), the increasing quantity of gas produced will determine the volume of the balloon to increase as well.