They already do. I'm not sure there's any legal benefit, requirements, or standing for being a civil rights organization though. Advocating the right of any group to do anything that may or may not be controversial, legal, illegal, or not yet recognized would be a civil rights organization. Unless someone can tell me otherwise, the phrase is a sound bite not a protected/enshrined class of advocacy groups.
Also remember, that because of legal regulations there are SEVERAL versions of the NRA. There's the NRA itself- the parent organization that conducts the safety classes, the public service announcements, and so on. There's also the NRA-ILA (Institute for Legislative Action) that does the lobbying, and possibly the judicial challenges. One of the reasons for keeping these separate both for us, and for them, is the tax exempt status of their donations.