It isn't prohibited by the constitution and was created as an independent body. You are correct that it wasn't specifically outlined by the constitution but that's an empty statement - the constitution included the allowance of the creation of agencies and laws outside of itself... that's the main power and strength of the constitution.
Your statement was equivalent to saying that Hacker News has no constitutional right to exist - it is equally vapid.
Maybe unsaid but the idea that Congress can create an organization that is not executed by the executive is the issue
The 10th amendment limits the federal government to the enumerated powers. So your argument that it "isn't prohibited" isn't how this works. You have to justify it under an enumerated power (which, you might well be able to do, though this is debatable).