The ACLU has two affiliated organizations; one allows contributors to take a tax deduction, and the other does not. The one that provides for tax deductible contributions does pro-bono litigation to protect civil rights, the one that does not is focused on lobbying for and against specific legislation.
http://www.aclu.org/donating-american-civil-liberties-union-and-aclu-foundation-what-difference
The ACLU comprises two separate corporate entities, the American Civil Liberties Union and the ACLU Foundation.... Both organizations support the ACLU's extensive litigation, communications and public education programs. However, certain important activities carried out by the ACLU cannot be supported with tax-deductible gifts. This is primarily in the area of lobbying, because federal law limits the amount that a tax-exempt organization like the ACLU Foundation can spend on lobbying.
I think that supporting specific candidates (electioneering) isn't allowed for non-profits regardless of whether tax deductions for gifts are allowed, but I am not 100% sure what the restrictions are. I am pretty sure that electioneering is even more restricted than is lobbying.
I'm not sure how AARP structures itself, but my guess is that AARP members would be pretty pissed if it were to engage in blatant electioneering. My guess is that NRA members wouldn't and don't care.