In Buckley v. Valeo (424 U.S. 1 (1976)), when the Federal Election Commission was stood up,
the original law was that the Speaker of the House, the President of the
Senate, and the US President would all get to appoint 2 FEC Commissioners each.
The US Supreme Court struck it down because, even though both the Executive and
the Legislative branches liked the law, it was a clear violation of Separation
of Powers because it gave Congress the ability to appoint people to an
exclusively Executive Branch agency.
Although the function of the
FEC was exclusively the purview of the Executive Branch, it had members
who were beholden to the Legislative Branch.
Under the Constitution,
'Officers of the United States' must be appointed by the President.
Btw, Congress probably can establish criteria and qualifications (such as
'must be a lawyer', or '50% of the Commissioners must be Democrats'),
which gives them some general power of who the President can appoint.
That issue has never been
decided in court, but it is pretty well established.
A second, separate issue that was decided in Buckley v.
Valeo had to do with campaign contributions and the 1st Amendment.
US Supreme Court found that
political contributions were a form of speech and thus covered by the 1st
Amendment.
The Court applied a balancing
test between 1st Amendment
guarantees of free speech and the compelling government
interest in fighting corruption and
bribery. They found:
Any limits on expenditures
by candidates are a violation of the 1st Amendment.
Limits on contributions can
be constitutional, but not if the limits are set too low.
Also, provisions that
require disclosure and reporting of campaign contribution are also
constitutional.
See Randall v. Sorrell (548 U.S. 230 (2006)), which reiterated this
decision.