Dunno. What is certain is that Trump can't appoint a new Director without Senate confirmation. And I'm not even certain about that. He may be able to appoint an acting Director for an indeterminate amount of time. In particular,
There are no statutory conditions on the President’s
authority to remove the FBI Director.
https://fas.org/sgp/crs/misc/R41850.pdfFrom the same cite, a President can nominate an acting Director. On the day after the death of long-time Director J. Edgar Hoover, L. Patrick Gray was appointed acting Director.
The Constitution was set up with checks and balances. But it requires the branches of government to do their duty. When you have a corrupt President of the same party that controls Congress, that may be asking too much.