It's what they don't say that matters.

From Media Matters:

Major gun violence prevention groups are upfront about their support for law-abiding citizens to be able to keep firearms for their own protection. Here's what the website of the Brady Campaign to Prevent Gun Violence says:

We believe that law-abiding citizens should be able to buy and keep firearms. And we believe there are sensible gun laws that we can and should insist upon when it comes to gun ownership.

And here's Mayors Against Illegal Guns:

We support the Second Amendment and the rights of citizens to own guns. We recognize that the vast majority of gun dealers and gun owners carefully follow the law. And we know that a policy that is appropriate for a small town in one region of the country is not necessarily appropriate for a big city in another region of the country.

Reading those quotes from the Brady Campaign and MAIG, I don't see anything that explicitly supports the right to keep firearms for self-defense.

The Brady Campaign says they "believe that law-abiding citizens should be able to buy and keep firearms." Buying and keeping a firearm is one thing. Being able to lawfully carry and/or use it for self-defense is another. For example, a modern day English subject can (under extraordinarily draconian restrictions) buy and keep a firearm. However the use of it in self-defense will result in a night in jail. And if you dare carry a firearm, you are automatically a criminal.

If you peruse the Brady Campaign's website, you won't find anything in support of self-defense in the home or outside of the home. On the contrary, you will find plenty against self-defense. For example, their opposition to HR 822 which would protect the right of self-defense across state lines and their opposition to Castle Doctrine which protects the right of self-defense in the home.

The story is no different for MAIG. Nothing on their website explicitly states that they support the right of self-defense inside or outside the home. MAIG is opposed to HR 822. MAIG founder Mike Bloomberg is also no fan of guns or gun owner's period.

So where is this upfront support for law-abiding citizens to be able to keep firearms for their own protection? Definitely not on their websites.