What would be a lighter-than-air gas that is also not flammable, and would be fairly easy to produce in a modern-style world?
Hot air. Cheap, readily available, nonflammable. The hotter, the liftier. You can replenish your supply anywhere you can breathe.
Gaseous water. Cheap, readily available, nonflammable. The hotter, the liftier. You can replenish your supply a lot of places. You will need a container that can withstand the steam.
Anhydrous ammonia. Dark horse candidate and never used as a lift gas to my knowledge. But definitely lighter than air and boiling at -33 C, a gas at most earthly temperatures. Caustic. Technically flammable but only at high oxygen levels so no explosion risk. Ammonia can be synthesized from N2.
If you can produce a thin rigid shell for your airship out of carbon or silicon, then you use the atmosphere itself as your lift gas by creating a vacuum inside the shell. Neal Stephenson’s Diamond Age used this principle to make light evacuated cells that were used for tiny drones and giant airships.