HOLY SHIT - I got going - scroll down to the bold for your answer…
I disagree with the assessment that the NRA is just a lobby arm for manufactures. Unless there is a total ban, they will just make arms that fit with in the legal parameters. Indeed there have been innovations and cheesy work arounds for more restrictive states. But anyway…
That said, yes and no. The NRA has the most real power, because they have a large voter base that is highly mobilized. While some people focus on the campaign contributions, direct contributions are generally paltry, though their PAC dollars are higher. So if you want something with similar power the answer is no.
Despite @JeffreyHayner’s assertion that they need to "get back to “teaching marksmanship, firearm and hunter safety and sanctioning competitions”, they are STILL really the only game in two for certifying instructors, and have the largest network of instructors. I am not 100% sure on how involved they are in competitions, I know they still are involved in some competitions. But as far as that goes, pick your preference and then see what is out there.
I do agree with @JeffreyHayner 's frustration with their “clouded… political bullshit”, although a lot of that comes directly from their “spokes people” who, as their disclaimers will show, “may or may not reflect the views of other individuals or organizations”. I suppose this is just playing to their base to a degree, but you will find people who fall under various labels as moderates, libertarians, socialists, and other flavors who still support hard line 2nd Amendment stances. They should be more inclusive and work to be less toxic. BUT, the reality is, shit like they are doing gets shared and goes viral. And it isn’t like their competition isn’t doing the same stuff on social media. I honestly think this is a symptom of a the bigger issue of social media and how we interact, but that is for another time.
At the same time, I disagree with some of the characterization of the NRA and supporters of the 2nd Amendment. They let a black, socialist who raps about revolution and civil rights, who supports things like BLM and Bernie Sanders, talk about the 2nd Amendment and how it relates to black rights for 42+ min (though of course people used pull quotes to push a narrative afterwards, but I agree with pretty much 100% of what Killer Mike says and his out look on rights and crime. It is long but worth a listen.)
TO ANSWER YOUR QUESTION:
Yes, there are other 2nd rights groups. A lot of them are more specific to a demographic.
Gun Owners of America is the 2nd largest group, but honestly, I don’t know WTF they other than take dues, provide “sky is falling” alerts, and give you an extra sticker for your truck. I could be wrong, but IMO they are amateur hour and don’t do much.
I have heard of the 2nd Amendment Foundation, but honestly I don’t know a lot about them.
Do note, I think the above two are MORE hard line than the NRA. YMMV.
Friends of the NRA - these are LOCAL groups who aren’t directly associated with the NRA, but they do things like raise money for training classes and hold competitions. This varies from state to state, locale to locale.
The Pink Pistols are a group dedicated to arming and training the LGBQT community. They focus more on training, and are a smaller group, but if you live in a large city they probably have a chapter.
Jews for the Preservation of Fire Arms is obviously from the Jewish perspective.
Black Gun Owners Association is from the black perspective.
National Shooting Sports Foundation - this focuses more on the various types of competition shooting sports
I know there is a liberal specific rights group, and I know there are some newer groups cropping up with socialist slants. Also locally some states have state specific groups.
All of these will have some local pull, or is a good place to find training and competitions for fun or serious. Good luck in your adventure.
I dunno - you do that and one of the editors might burn it down. It certainly is fairly popular.