What is the role of carbon dioxide in a soda-acid fire extinguisher?

You turn it upside down or strike a plunger (depends on model), to release sulfuric acid into the water containing sodium bicarbonate, resulting in carbon dioxide gas pressure. The pressurized gas expels the water/acid solution out the hose.

Of course, under most US and Canadian fire codes, this type of fire extinguisher has been obsolete for many years and must be removed from service immediately, if any are found.