We need to distinguish between the different sorts of non-profits, what they can do, and how effective they can be; I'll use the NRA as a concrete example because I know it well and many in this fight had noticed how very well it fights its own battles ... plus now it's filed an amicus curiae brief supporting an ACLU lawsuit. (One of the very encouraging aspects of this battle is that it's very much not breaking down by the usual lines.)

So the first thing to remember is that "non-profit" or 501(c) does not equal charity, although a normal charity will be a non-profit, and if they jump through the proper hoops and the IRS doesn't politicize the process they'll be a 501(c)(3). Note that the IRS has very much not stopped its stonewalling, and we can expect anti-spying non-profits to get the same treatment that "TEA Party" et. al. groups have gotten ... something to think about if you don't take to heard the advice written in blood about "First they came for the...."

So:

501(c)(3) organizations like the NRA proper or the EFF can do a variety of things including lawsuits, but when it comes to politics I think they can only "educate their members". Of course, if you use a web site as part of the latter....

501(c)(4) organizations like the NRA's Institute for Legislative Action (ILA) can do explicitly political things. Exactly how that breaks down is complicated, but they can do stuff like directly lobby legislators as long as that's not most of what they do. One very powerful thing they do is rate and endorse politicians and send out voter's guides before elections, little "just the facts, Mam" postcards that show how the relevant politicians voted on issues their members care about.

But there's a critical caveat here: this only works for the NRA/gun owners (NRA members inform their non-member friends of what they learn) because there are so very many of us, we have a very strong grass roots that's by no means tied to the NRA proper (as some politicians have learned to their dismay), and we vote.

Look at the Total Recall of 2 Colorado state legislators including the Senate President due to their prominent roles in passing invidious gun control legislation. This was done from the ground up, one of the 2 efforts was started by 2-3 plumbers who'd never done this sort of thing before. The NRA only spent ~ $350K in the two efforts, the good guys were, last time I checked, outspent 8:1 by Bloomberg and the usual suspects, the MSM was on the anti-gun side, and the election laws had been ... curiously changed to make voting fraud very easy and it was brazen. Yet given all that, two career politicians will very soon be spending more time with their families.

Which gets to our particular problem, which is that we don't have that sort of grassroots, not will we if the NRA/gunowners are any guide before the national security state commits enough notorious atrocities, see here for more on this theme: https://news.ycombinator.com/item?id=6185918

Now to the more political organizations: the NRA of course has a PAC, the Political Victory Fund, but ever blessed post-Watergate campaign finance "reform" ensures the money they can give is purely symbolic. I.e. it sends a signal to gun voters, but little else.

Then there are the newer 527 Super Pac organizations. I haven't studied them closely, but as I understand it they can do just about everything but "coordinate" their efforts with a political campaign. But getting back to my above point about voting, unless you have a strong base of potential voters they can spend zillions of dollars without having anything to show for it. For the latest notorious example of this, see how in 2012 the Republican Party national level establishment nominated a Northeast liberal politician for president, establishment organizations like Karl Rove's American Crossroads spend 100s of millions, hismore than $100 million, and had nothing to show for it.

A political party that despises its base is just not going to do well.

We should take that lesson to heart and not "despise" any who agree with our cause unless they're utterly toxic to it (I mean NAMBLA, Stormfront, and ALA levels of toxic).