This is why I would have translated it to "seek" which doesn't need "for" after it.
Is there a difference between "Sie suchen Holz" and "Sie suchen nach Holz"? I think they are using both in the excercises and I haven't found out what the difference between the two of them is (if there is a difference, that is).
In general, there is no difference. But when using 'nach', the objects case changes. Without: accusative, with: dative.
As a native English speaker, I find this worded oddly. Would 'They are looking for wood' or 'They are trying to find wood' be accepted?
"They are looking for wood" ought to be; try it, and report it if it isn't.