Theoretically, Biden could do it and not be prosecuted.
But if he ordered a member of the military to do it, they are required to refuse illegal orders. I don’t know the rules about illegal orders but I bet this would fall under that. At the same time, the President can pardon people convicted in military court so that’s not much of a deterrent.
Similarly if he ordered a civilian (say, CIA) to assassinate Trump, that person could be tried. But again, the President’s pardon power makes federal charges not much of a threat.
BUT — the President cannot grant pardons for convictions in state courts. So anyone involved would be in trouble if it happened in a US state. And if the Supreme Court did not make the President immune from state-level prosecution, Biden could be tried for being involved… but it seems unlikely that they would go for “the President is immune from federal prosecution but not state prosecution.”
Of course, all this show how insane and dangerous the idea of Presidential immunity is. It’s a terrible idea.
WadeTheWizard