So let's say you have a Dyson sphere that is 5 kilometers thick . Now, it isn't solid. It's meant to be inhabited. There's the inner shell which is completely covered with solar panels and any other manner of power generation facing the star. It's 500 meters thick. There's the outer shell which again is 500 meters thick, but is covered in armored plating, defensive weaponry, and the occasional hanger door. Between these two shells is an empty 4 kilometer area that is FILLED with breathable air. People living in the sphere would inhabit centrifugally spinning towers that act as support columns between the shells, but I digress. Where would one go about getting all that air? This is assuming that the creators of the sphere are humanesque (more or less), and require the same air we do.
You Must Find a Source Outside Our Solar System
Given the size of a Dyson sphere, it's fair to say that you need a LOT of gas.
An idealized Dyson Sphere is large enough to contain planets such as Earth and, per Wikipedia, would be about 1AU in size. Given that such a sphere CONTAINS our planets and their orbits, it's intuitive that you could not possibly harvest an amount of elemental oxygen necessary to create a breathable atmosphere.
Let's say though that you wanted only to build your Dyson Sphere around the sun, leaving the planets cold (or transmitting directed energy to them from your Dyson Sphere). That's much smaller. Your challenge is that even Jupiter - our largest planet - could fit inside the sun 1,000 times over. In fact, the sun comprises 99.8% of the mass of the solar system itself!
In that scenario, if you wanted to fill it with air from this solar system you are trying to build a sphere around a body with 99.8% of the solar system's mass that is 4km thick and fill it with breathable air from the remaining 0.2% of available mass in the solar system, only about 5% of which is actually oxygen. That 5% number also needs to contribute to at least 19.5% of the atmosphere of your creation.
Long/short - There is not enough elemental oxygen in the solar system to allow a Dyson sphere with a 4km atmosphere of breathable air. You must obtain your oxygen from another solar system; quite likely MANY other solar systems.
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.
So the good news is that if you have the capability to construct a Dyson sphere, you have the capability of supplying it with resources.
The sheer amount of matter needed is huge, and you'd need to steal the planets and other debris from many other stars to get enough. And since you're stealing all that stuff you can get their gasses too.
If you don't like the idea of moving dozens of solar systems worth of planets across light years of space, the other option is to make an energy to matter converter. Then you just need a whole lot of energy, which can be provided by the sun.
Best course would be to collect the asteroid belt, kupiter belt, and any other stray rocks that are floating around, turning them into a Dyson swarm with as many energy collectors as you can put up, and start feeding it all to an e->m engine. Use that to expand your collectors until they form a Dyson ring, and then enclose the globe.
This would be an exponential process, since the more you enclose the more energy you have to work with.
At this point you've got your inner shell, and you should have all the energy you need to finish the outer shell and then start making atmosphere to fill it.