The builders would plunder the star's planetary system (and neighbouring systems, as needed) for raw materials when building the Dyson shell.

Without knowing the exact dimensions it's impossible to say how much gas is needed to fill the inhabited gap, but assuming "air" here means a mix of gases – nitrogen, oxygen, carbon dioxide, etc – comparable to Earth's atmosphere, these gases can be harvested from the atmospheres of gas giants and ice giants and the crusts of terrestrial planets:

Electrolysis, photolysis and other methods will be needed to wrest the desired gases from their bonds (such as nitrogen from ammonia, oxygen from ice water and iron oxide), but if you're building something the size of a planetary system that should be well within your ability.

Bonus: The builders would end up with a lot of free hydrogen and helium, which they could put through an artificial neucleosynthesis (fusion) process, to produce needed elements.

The builders would plunder the star's planetary system (and neighbouring systems, as needed) for raw materials when building the Dyson shell.

Without knowing the exact dimensions it's impossible to say how much gas is needed to fill the inhabited gap, but assuming "air" here means a mix of gases – nitrogen, oxygen, carbon dioxide, etc – comparable to Earth's atmosphere, these gases can be harvested from the atmospheres and satellites of gas giants and ice giants and the crusts of terrestrial planets:

Electrolysis, photolysis and other methods will be needed to wrest the desired gases from their bonds (such as nitrogen from ammonia, oxygen from ice water and iron oxide), but if you're building something the size of a planetary system that should be well within your ability.

Bonus: The builders would end up with a lot of free hydrogen and helium, which they could put through an artificial neucleosynthesis (fusion) process, to produce needed elements.