Desmume had internet connectivity at one point but it was removed because Nintendo could tell when traffic was coming from it (not fully sure how) and take action against those connections.
I haven't played B/W enough to know much about the battle recording features but if it's like HG/SS couldn't he have done the match on a real DS, saved it, and played it back while running on an emulator?
What do you mean by WIFI in emulators, and what do the other claim to know when they say NO ?
I know at least one emulator that successfully uses WIFI with the Nintendo DS, and that is the mighty ZXDS.
In fact with ZXDS you can connect to the internet and download games from some legal (partially ) rom sites, as well as playing two players Spectrum games via local WIFI multiplayer.
So it's possible I would say, as some brilliant emulator author already implemented it.
Never mind, I reread your post and now I know you mean WIFI for a DS emulator for the PC.