A trust or corporation as a legal entity for the purchase of NFA items isn't going away.
That EO wouldn't hold up to even a half assed legal challenge.
At best they might require the person receiving the NFA on behalf of the trust or corporation to undergo a NICS.
Requiring all trustees or all corporate officers to undergo a background check will not stand up to a legal challenge either.
This shit has been talked about for years (as long as I can remember and I've been doing the trust thing since before Obama) all the NFA lawyers have said it would not stand.
Obama can sign anything he wants, making it stick past when the ink dries is a whole other story.