I think that the President of the United States is entirely within his rights to shut down such an organization. He has the power to do so, and there is no reason why he should not. Organizing for Action is a group that is not without its problems, and needs to go.
Obama should keep the Organizing for Action because it is a non profit organization where donations are accepted by the people, and corporations, keeping it open for citizens to make decisions. This organization is made up from fifty different organizations from the states. They will carry meetings concerning gun control and immigration. It is not up to the government, it is up to the people. Our decisions will count.
No, the First Amendment protects free speech and freedom of assembly in America, especially by a non-governmental entity such as Organizing for Action. The founders of this country supported free discourse and acknowledged that sometimes, the most unpopular opinions still need to be heard in order for there to be a functioning democracy.
Organizing for Action is an organization independent of the government. He can't shut it down. It's just not in his power to do so. All he can do is ask, and if the organization wants to continue to function, it will. It's just an advocacy group, and there shouldn't be much controversy about that.