If Biden / Trump / a hypothetical USA president elected in the near-future wanted to abolish the FBI, how could he?
Can the president do this alone or would they need a Senate vote too?
How could an American president abolish the FBI?
The president can't abolish the FBI without a law, which law requires a majority vote from both the House and Senate and the signature of the president.
Furthermore, the FBI has exclusive enforcement for some crimes. Each of those crimes would need to be reassigned to another agency or they become unenforceable.
The president can reduce the effectiveness of the FBI through appointments or lack thereof; but, until the law is changed, any future president could reverse the prior decision.
Presidential Reorganization Authority: History, Recent Initiatives, and Options for Congress, December 11, 2012
Between 1932 and 1981, Congress periodically delegated authority to the President that allowed him to develop plans for reorganization of portions of the federal government and to present those plans to Congress for consideration under special parliamentary procedures. Under these procedures, the President’s plan would go into effect unless one or both houses of Congress passed a resolution rejecting the plan, a process referred to as a “legislative veto.” This process favored the President’s plan because, absent congressional action, the default was for the plan to go into effect. In contrast to the regular legislative process, the burden of action under these versions of presidential reorganization authority rested with opponents rather than supporters of the plan. In 1984, the mechanism was amended to require Congress to act affirmatively in order for a plan to go into force. This arguably shifted the balance of power to Congress. The authority expired at the end of 1984 and therefore has not been available to the President since then. [Emboldening added.]
They could try.
The Federal Bureau of Investigation evolved out of agencies created by the Department of Justice, and Office of the President of the United States. The FBI's direct precursor organization was created by Executive Order 6166 on June 10th, 1933.
As such, a sitting President could, if they had the cooperation of their Attorney General, attempt to dissolve the agency, reassign all mandated responsibilities to other agencies (or revert them to the DoJ itself). The stand-down process would be extensive and involve a massive commitment of resources, which creates a number of potential failure points.
As pointed out in the comments and elsewhere, the reaction would be an immediate lawsuit that would no doubt find itself express-lane'd to SCOTUS. Then it becomes a battle:
Minutiae of precedent surrounding Congress' authority to designate funds and the limits on the President's power to redirect those funds.
vs.
The "hand that giveth" principle.
Team FBI argues that funds appropriated for the FBI are legally equivalent to funds appropriated for distribution to the states, and that 28 U.S.C. Section 531 indirectly causes the existence of the FBI by implication.
Team President argues that funds allocated for distribution to states by Congress are fundamentally different things, that Train vs. City of New York held that the President could not impound funds designated to be sent to other entities. Whereas the FBI, as part of the Department of Justice, is in fact and in law, subordinate to the Executive Branch and therefore those funds have not been withheld from their proper beneficiary. They attack the 'defined into existence' argument by pointing out that nowhere in 28 U.S.C. Section 531 do the words "There shall be a Federal Bureau of Investigation" appear, and that this section should be interpreted as merely assigning such a department, if one exists, to DoJ. It's not the strongest counterattack, but flimsier arguments have won the day and this one has going for it, the "hand that giveth" principle, which is, fully stated: "The hand that giveth, may taketh away," and is the concept in political philosophy that it is impossible for anyone possessed of legitimate authority to delegate that authority away irrevocably.
Team FBI's counter attack is to point out that "the hand that giveth" does not preclude other hands deciding to also giveth, and that the 'givething' becomes a logical OR statement.
The outcome of this depends on the constitution of SCOTUS and the Congress at the time when it is attempted. There are pathways for the President to succeed in the attempt, but they are far from certain and of course the Congress can remedy any fault in "defined into existence by implication" by adding "There shall be a Federal Bureau of Investigation" to 28 U.S.C., and if they were strongly opposed to the President's move, they could probably get this done inside an hour or so.
Finally, the President has strong legal power to limit the actions of the FBI, regardless. The notion that the Department of Justice (the parent org of the FBI) is separate from the President is a normative one: it is so because people generally respect that it ought to be. That separation appears nowhere in law.