Are all quadrupeds edible?
Edible by whom? Lets assume humans.

Yes all quadrpeds are edible though you may not want to eat every part.

For example, the scent glands of a skunk, or the quills of a porcupine are repulsive or impossible to eat. Also, some organs, like the liver of a polar bear, which accumulates vitamin A to toxic (to humans) levels.

It's impossible to say why this is so, since we can't prove why evolution occurs one way or another. What you suggest about mobility is reasonable, but not universal, since the poisonous puffer fish is able to move.