Yes, the NRA supports this bill. Not only have they said it many times (for instance, here at their factsheet on HR 2640 and in this article), but the Congressional discussion includes praise of how the Congress is working with the NRA on this bill.
I am a member of the NRA, and I don't support this bill. I called the NRA to talk about it, and the first person that I spoke with was the least knowledgeable individual on firearms issues that I have ever spoke with at any firearms organization (and she had a very odd accent which I would be unsurprised to find was Indian or Pakistani).
A bit frustrated, I called another number and found someone who at least had good knowledge of firearms issues. He said that if the NRA thought that any bad items might be attached to the bill that they would then withdraw their support. (A risible statement in my opinion since by that point I think that the bill would be unstoppable.)
The community is definitely split. Clayton Cramer supports HR 2640.