What gas law should we use to show that the volume of a gas is directly proportional to the amount of gas?
That would be Avogadro's law, 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
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.