The Fed Board of Governors are appointed by presidents for 14 year terms and are not directly responsible to the Executive after that. He can choose who is the Chair, Vice Chair and one other role, but that is from the existing members. He can also remove for cause. Currently, most of the board was appointed by Trump, iirc.
Janet Yellen used to be the Fed Chair. They are supposed to be apolitical, but like anyone in those roles, they definitely have personal bias. Ethically, she should never have taken that role as Treasury Secretary because it is overtly political.