Very Unlikely. While there should be no scarcity of lift gasses (Hydrogen is damn near everywhere and assuming that it's a fundamental building block of life for your world, it should be quite easily found in any water), there's a problem with extracting it that you encounter.
Electricity has not been discovered because of the lack of metals so there is no aluminum either (although it is present).
So first of all, the word electricity was first used in 1646, though the phenomena was known 2750 BC, when Egyptians described electric fish native to the Nile as "Thunderers of the Nile" (although the relationship between Lightning, Thunder, and Electricity wasn't proven until the experiments of Benjamin "I'll fly a kite in a thunderstorm and see what happens and then help found a nation" Franklin made the discovery. By comparison, Aluminum wasn't discovered until 1825 although you are right about the scarcity, as it was extremely expensive. Napoleon III (not the short but not really Emperor of France, but a couple successors later) had a full dinner set made of Aluminum that was used only for the fanciest of dinners... when the golden dinner sets just wouldn't impress enough. And the United States capped the Washington Monument with a single Ingot of Aluminum to symbolize the wealth of the new nation. A method to synthesize aluminum in such a quantity that we can drink soda from cans made of the stuff for less than a dollar wouldn't be invented until 1886... and by this point Thomas Edison had already invented the lightbulb, was illuminating cities with DC Power in the U.S. and Europe, and had this uppity Hungarian immigrant named Tesla quit because he couldn't take a joke and invent AC power to compete... 1885 was the start of The War of the Currents!
All of this is a problem because the best way to make a lighter than atmosphere gas, hydrogen, would be to take some water, run electric current through the water, and separate the hydrogen gas from the oxygen gas in such a way that the highly reactive O2 gas, doesn't meet with the highly flammable H2 gas and... well, when I ask you what the first thing that comes to mind when I say the name "Hindenburg," I'm going to go out on a limb and say your answer will not mention the German President from 1925 to 1934 Paul von Hindenburg beyond the passing mention that the more famous one was named in his honor.
The only other lift gas available is Helium. Which while it's cheap enough to fill children's party balloons, or was... prices have doubled in the past decade. Unlike Hydrogen, Helium cannot be synthesized... well... not without nuclear physics involved or a particle accelerator and that's not happening if we're stuck on a possible internal combustion engine on a non-metal airship... as it is lighter than air, it has a tendency to wander away from earth... but unlike Hydrogen, Helium is a noble gas... which means you can only have it in a pure state. Almost all of the world's supply is mined from natural gas found in mines under the United States (90%) to be exact. The remaining 10% is produced by Canada, Russia, and Poland (It's at this point if your still asking "Why didn't the Swastika-emblazoned Hindenburg use non-reactive Helium instead of the explosively reactive Hydrogen, don't the Germans understand Chemistry?" I must question the validity of any passing grade you received in world history.). Hydrogen would be more likely your lifting gas because it's far more plentiful and renewable. Hydrogen provides more bang for your buck (Don't you dare tell me it's "Too Soon" for that joke, I stand by it.).
That said, arguably the World's First Aircraft Carrier was fielded by the Union forces in the Civil War, a converted river barge that could launch and recover a single hot air balloon teathered to it's deck with a telegram wire to provide battle field intel. Hot air rises, no matter the gaseous mixture. There was even a hypothesis that the Nazca Lines were drawn by pre-Columbian natives in Peru using hot air balloons, although this theory seems to be debunked by a dedicate team of archeologist who were able to produce similar designs in mere days with some planning and coordination. Still, it's not the most ludicrous theory out there regarding the Nazca structures. Looking at you, Giorgio "I'm not saying its aliens... but aliens" Tsoukalos!
All this is to say that a society that never had a bronze age would likely not have the capability to develop an internal combustion engine, which would require metals, nor electricity to extract hydrogen, which also requires metals, or gas mining for Helium (which requires metals). I don't think this idea passes a smell test. Unless your world is... nope... nope... not going to say it... I've already made a reference to Ancient Aliens and because of course someone who is a meme for trying to say every human achievement ever done was because of aliens has to have a name I cannot spell... Of course he does...