Microorganism-powered aerial flight is plausible, but with caveats.
As you point out in your question, hydrogen is the best lifting gas we have. So this becomes a question of "What's the most efficient biological process that produces H$_2$?" The answer, of course, is algae.
Normally, algae get their energy from photosynthesis- taking in sunlight, water, and carbon dioxide to produce ATP, complex sugars, and oxygen. However, under the right conditions (mumble mumble sulfur-limitation heterocysts mumble) some algae will switch to a metabolic state of "anaerobic oxygenic photosynthesis". In this state, the oxygen produced by photosynthesis is used by the cell's own respiration, producing an anoxic environment, which in turn triggers the production of hydrogen gas.
What this means is that algae can produce H$_2$ gas almost directly from protons. Even better, we can collect it and are already well on our way to making it a cost-effective replacement for fossil fuels. Clean energy in our lifetime? Yes please.
However, that's not enough to answer the question, which asks about the rate of H$_2$ production. In 2001, a company built a 500 liter bioreactor that could produce an astounding 1 liter of hydrogen gas per hour. With that kind of potential, our balloon would need luck to even start inflating. However, that was 2001, and the first year the company started. At that time, they calculated a theoretical maximum of 20 grams of hydrogen per day- about 10 liters per hour. In 2004, a review came out that posited a maximum of 5.45 kg of H$_2$ per square meter per year. That's a rate of ~7 liters per hour- still a bit too slow. In 2011, we multiplied that rate by 5 times by creating biohybrid photosystems that use platinum nanoparticles. In 2013, we managed to do even better and increase our efficiency 4x by modifying the chlorophyll antennae, and that's since been pushed to 13x. So our current rate of H$_2$ production is about 450 liters per hour! Of course, this is an idealized maximum efficiency and we haven't yet managed it on a large scale.
So what does that mean for our balloon? In a world where people rely on balloons like this, I'm going to assume that they're operating pretty close to maximum efficiency, perhaps 400 liters per hour per square meter. Of course, there may be problems with this, but it's a decent estimate to start with. From skydrifters.com, we learn that an average hot air balloon weighs 800 pounds. Since we're traveling and trading, let's call it 500 kg total. Lifting 500 kg with hydrogen gas requires ~40 kg H$_2$. At normal air pressure and temperature, this occupies a volume of 450,000 liters. Thus, our balloon will take approximately 41 days to fill under its own power. That's going to be hard to pull off.
However, this calculation shows that algae can indeed produce enough gas to lift a balloon, and it'd certainly be an eco-friendly way to travel. It also allows for maneuverability in the air, and essentially permanent air travel. Once the balloon goes up, the algae can draw their CO$_2$ directly from the air and the balloon will be powered by light alone. Additionally, it's quite possible that towns and cities would start to farm fields of algae so that filling up at a city is quite easy and would entice ballooners to visit.
In the air, such a balloon would ascend normally as the algae produce hydrogen gas. Additionally, H$_2$ gas compresses quite nicely, and it might make sense for ballooners to keep a compressor on board to capture any excess H$_2$ produced for a quick burst if necessary. Descending is the easy part: the simplest would be venting the H$_2$ or compressing it for later. You could also spray the inside of the balloon with some kind of inorganic sulfur, which would shut down the hijacked photosynthesis pathways temporarily, or add oxygen, which would destroy some of the hydrogenase enzyme.
The mental image I have of this system is a very gross, quite large clear balloon. The outer membrane would be made of plastic wrap or some other impermeable lightweight clear solid, and there would be layers of algae directly inside. Any H$_2$ produced would fill the balloon and contribute to lift, displacing any denser air in the meantime. Since the microbes live best in an anoxic environment, this wouldn't even be a problem. Maintenence would essentially consist of replacing nutrients and removing dead cells from the inside, which would probably be done when on the ground but could be done in the air if one can hold their breath long enough.