Cells need oxygen to function and clearly the cells in a zombie are at least partly functional. In the Walking Dead, do zombies need oxygen? Can they breathe? If they don't have functional respiratory or circulatory systems, do they absorb some oxygen through exposed tissue?
No. In pretty much any mythology (at least where zombies are dead, not living 'infected') zombies do not need to breathe at all. There is no explanation given for how they function without oxygen, nor is there an explanation for where the energy they use comes from.
Zombies have been seen to operate just fine underwater for extended periods, and zombie heads are still animate (unless the brain is destroyed) even without lungs. It is probable that a zombie could function in a vacuum, unless they were frozen stiff.
This is, frankly, a conceit of the genre. There's no real explanation, and I don't think there can be one. We do, however, know that they don't need oxygen to function.
Why do you assume their cells to be functional? In fact, in every single zombie representation I am aware of zombies don't heal. If their cells were working, I'd expect them to have normal healing powers but generally, when you cut a zombie, the cut stays open and you get those wonderfully gruesome images that so enhance our viewing pleasure.
So, while no explanation is given in any canon I know, the fact that they don't heal would point to their being animate but non-living tissue (whatever that may mean) and as such I would not expect them to need or use oxygen, no.