Short answer: Volcanic eruptions or nitric acid breathing by animals/funghi might be the best way to go.
I'm assuming you mean a pH of 0-2, this is would be really acidic. For this, a strong acid gas is needed. It must be a gas to cause acid rain and it must be created in the atmosphere/on land, not in the oceans because there it would react immediately with the water, leading to acidic oceans but not to acid rain. In the atmosphere the acidic gas would react with water and cause acidic rain. There are 3 common (and a lot less common) acid gasses.
Sulphuric acid (H2SO4) is quite common in volcanic eruptions leading to acid rain here on earth from time to time. Sulfur doesn't really play a major role in land biology (quantitatively), so there is no good reason why it would be mass-produced by land organisms.
Nitric acid (HNO3) contains nitrogen. Nitrogen is way too valuable for any primary producer (plants, some bacteria) to be emitted into the atmosphere, but heterotrophic organisms (animals,funghi) need to get rid of nitrogen. Animals usually do this by excreting it in a solution (e.g. urine), but I guess it would not be completely implausible to get rid of it by breathing out HNO3 just like they get rid of carbon with breathing out CO2.
Hydrochloric acid (HCl) contains chlorine. This is generally abundant in the ocean, but it requires a lot of enery to create HCl from it. I could imagine some plants at the coast (some sort of mangroves) that use some sort of biogenic electrolysis to get minerals from the seawater and create HCl in the process as waste-product. Although this would hardly be enough to create enough of it to alter the atmosphere significantly.
All of this gases are high-energy compounds, so organisms need a good reason to use energy on it, e.g. because they need to get rid of a waste-product, just like oxygen. The most plausible is probably the sulfuric acid by volcanic eruptions. The least implausible biogenic explanation is probably the breathing of nitric acids by heterotrophs. While the volcanic eruptions are a relatively solid explanation, the nitric acid breathing is speculative.