So why is the first "aus" needed?
Like the other answers noted, the preposition is necessary to denote the object. There is also "ausziehen" without a preposition, but that uses the object in accusative case, and has a different meaning: "Wir ziehen den Pullover aus" - "We take off the pullover". It doesn't make sense to use this form for an apartment ("Wir ziehen die Wohnung aus.") :-)
I.e. is it correct to say... "Wir ziehen der Wohnung aus."
No. That would be as incorrect as the English "We move out the apartment".
However, you can omit the second "aus":
"Wir ziehen aus der Wohnung (aus)."
- "We move (out) from the apartment." / "We move (out) out of the apartment."
In this case the verb "ziehen" is understood as a short form for "umziehen", although it also has the meaning of being part of a procession ("umherziehen", for which I don't know a direct translation). You'd more customarily say something like
"Wir ziehen aus der Wohnung in eine andere (um)."
- "We move from the apartment into another."