Really? Do they actually say this, or do you just believe "they will say this?"
I don't believe I have ever heard NRA make an explicit statement opposed.
However, Zack991's comment lines up well with I know.
1. NRA is heavily supportive of CC instructors.
2. It is an article of faith among many CCers that you never, ever let your gun show in public. Element of surprise, and don't scare the soccer moms, and don't get police calls against yourself.
3. Police opposition to OC is so common as to be almost axiomatic.
4. NRA is seems heavily supportive of police or connected to police. It seems like I've seen a number of NRA publications or releases with some reference to training cop instructors, or making some community service activity in connection with police. It is almost as if the NRA puts police on a pedestal. Or, capitalizes on public affections for police. (Unless the police are confiscating guns after a hurricane.)
Also, you should have seen the fuss a couple years ago when they sent out an NRA-ILA alert about CC in alcohol-serving restaurants. They basically said carry was a no-no in such places, or something that omitted the perfect legality of OC in such businesses. When VA OCers pointed out the error, their response was that it was not an OC issue. Huh? How about just an issue of getting things right. Eventually they changed the e-mail alert.
In any event, I don't see NRA being openly supportive of OC anytime soon unless the political winds of the gun world make it beneficial to them.