I have absolutely no idea. 
I guess if they could possess trees, which presumably don't have souls, it would be easy to possess a souless husk? Or is the soul or connection to the Fade what makes it even possible? (...)
First of all, I wouldn't say that plants don't have souls. For what we know, the definition of mortal being with a soul is likely much too rigid - remember Lady of the Forest? I think she's stated to not come from across the Veil, but from the forest - wouldn't it make her pretty much the forest's soul? And if a forest can have a soul, why not a tree?
Also, spirits generally possess things that are - or at least were - alive. Possessed people, plants, animals... there are ash wraiths but they don't seem to have bodies as much as emulate them with their powers. Still, if they can possess a corpse, I believe a darkspawn could work too UNLESS Taint is actually a demon-repellant to some extent. Still, as fhs33721 says, we see possessed wardens... so it seems that the answer would be yes, there don't seem to be any solid rule that would make darkspawn impossible to possess. They may be of limited use and/or not very attractive but almost certainly possessable.
Although there is one aspect that just came to mind - perhaps a demon possessing a full-fledged living darkspawn would just be indistinguishable from normal darkspawn? If the Taint can ovverride human's reason, it's not impossible that it would "enslave" a demon the same way, making it just another darkspawn in the horde... Though I highly doubt that the taint would actually affect the demon itself. Saved Archdemon soul is supposedly NOT tainted. A demon freed after its darkspawn dies would likely be free of the taint too.