Sure it is. The Fed doesn't get any funding from Congress. What influence can they exert?
Greenspan offered to step aside to let W appoint his own Chairman (didn’t happen). It’s not law but Greenspan thought the president should have the Chairman he wants.
He didn't make the same offer to Clinton. It's not customary for the Fed Chair to resign on change of president in keeping with the independence of the Fed from the Executive. And even if she did resign immediately on day one, does Trump or any of his people know Tarullo, Powell, Fisher, or Brainard well enough to know if they would be any better?
Congress can call the Chairman for testimony. They can also prevent the Chairman from getting like minded fellow governors. So can the president. If worse comes to worse they can change the law that applies to the Fed.
I’ll also repeat that the actions of the Treasury can have a big influence on what the Fed does, or can do.
Like I said, they aren’t all that independent. They are mainly there to handle operations, do what the politicians want them to do and to take any blame that politicians wish to give them.