You can to! Steve at Cookshill does get them from time to time. You need to be ready when he has them. He sends huge amounts of material to the States so can advise if there is likely to be any problems. Woodcock is the most difficult of birds to skin, Its skin is like tissue paper, but Steve does produce skins.

We have woodcock locally, but as I gave up shooting some years ago (personal reasons - I still support your right to go shooting), I can't get you any myself. I'll ask those I know who do shoot if they have any wings. I may be able to get hold of a pair for you if you would like.
Cheers,
A.