Leave your moons in peace, harvest some nearby stars
As with all of Dyson's inventions, we'll have to hand-wave how it has to be produced anyway..
Any moon, or even planet Earth would be irrelevant, you may as well preserve it
Volume of a sphere:
V = 4/3 πr³
Volume of a hollow sphere:
V = 4/3 πrOuter³ - 4/3 πrInner³
Where rOuter-rInner = thickness of your Dyson sphere. Now suppose you'd rotate your Dyson sphere, the hull would need to provide support, not mass. In order to get a safe Dyson sphere, and a minimum of one km thickness (1000m) will do, with a density of say 3gr/cm (part is metal) you'll still need a giant amount of matter, when your Dyson sphere would be a solid construct (not "sparse")
Suppose you'd take the Earth orbit as a guideline for its size (less hot and more convenient would be Mars, so take Earth's orbit as a minimal radius r1) you would get
1 AE = 149 597 870 700 meters radius, which yields a total volume of 4.2e34 m³
For the Dyson hull, subtract like in above formula. You'd still need 8.43e26 m³ which is a few orders less of course, but assuming we need an average density of 3.0gr/cm³, you'd still need a mass of 2.531e+33 grams according to my calculation. To build that thing, you'd need 2.531e+30 kilograms of material.
The total Solar system including the sun has a mass of 2.1e+30 kg.
The sun itself has a mass of 1.9e+30 kg, so if you want to keep it, you're left with 0.2e30 kg of building material.
Conclusion: if you could harvest ALL MASS in the solar system except the sun, you'd have less than 10% of the material required.. so you'll have to set up some interstellar transports anyway, to import the material required !
Make the Dyson hull as thin as technologically feasible..
Say, you repeat the calculation for only 100 meters thickness, the required mass would be in the order of 10e+29 kg, still more than a million times the moon mass. But a hull weighing 10e+29 kg could (theoretically) be built by shredding our gas giants Jupiter and Saturn.
Solid or sparse, Dyson shell or Dyson swarm
A Dyson sphere harvests energy of a sun, that is its intended purpose. Inhabitable Dyson spheres are fiction.
A solid shell was first named by Dyson, but after that, he ensured his public that a solid sphere could never be constructed. The shape of the megastructure would be rather look like a swarm, like
The two differ in the way each "solar cell" would move. In the left case, everything is in orbit, only harvesting the energy. In the right case, propulsion is needed, which can be provided by the solar radiation itself. On the right, each satellite is stationary in space.
Material will be less of a problem, but you'll have to crush some moons to build it. The location - to start - could be between Mars and Jupiter, the asteroid belt, creating an Dyson ring from available material. Mars' moons Deimos and Pheibos can crushed first. Then some of Jupiter's moons.. the ring will take shape in a few centuries..
Notice that the Wiki-topic does not mention materials or "metal". When I proposed to use gas giants and stars to gather the mass needed, I ignored the fact that material for a solid Dyson sphere should be solid, or solidified, in some way. You could hand-wave that, but this is a science based topic.
To gather the energy, you'll need metal parts
How much metal would be needed and where harvest that.. Suppose you'd like to have cables connect each spoke of the above Dyson swarm of solar cells, gathering the energy.
A radius of 1AE, circumference will be
C = 2πr = 2 * 149597870700 * 3.14 = 939 951 143 167 meters cable per spoke
..suppose you'd need a cable of 1cm in diameter, that is 0.0003 m2, you'd need
V = πr2 * C = 0.0003 * 939 951 143 167 = 281 985 342 m³ metal per spoke.
Using iron, amounting to 7873 kg/m3 times that amount when it's iron: 2 220 070 605 tons. The Earth's yearly total production of iron in 2015 was about half of that amount, let's go for it.. we can do it now.
In the (far) future to construct a true, solid inhabitable Dyson shell like Star Trek shows us, like a giant armored inhabited sphere around a sun, humans could consider to go harvest iron in a supernova.. in a certain stage of a supernova, a massive amount of iron will be produced. Maybe the supernova could be stopped just before it occurs ? You'd have a giant solar core consisting of iron.