The president has some constitutional powers that cannot be removed without amending the constitution, which requires the agreement of state legislatures, so it cannot be done by congress alone.
Statutory law explicitly grants to the president the power to make certain decisions. This discretion can be removed from the president by amending the relevant statute. This can be done without the president's consent by overriding the president's veto.
The president also has implicit discretion, for example if a statute fails to specify whether it applies in certain cases, or when it is ambiguous. Since these things concern statutory law, however, they can also be resolved by amending the relevant statute or statutes.
So really the only powers that cannot be removed from the president are those specified in the constitution.
Could they effectively exclude him from all decision making if they wished so?
Most, but not all. Congress could not remove the president from the process of political appointment or of international diplomacy. They could not remove the president from command of the military.
They cannot remove the power of the executive, and because it would be impossible to create a body of statutory law that unambiguously requires specific actions in every instance, there will always be some implicit discretionary power for the president to exercise. Congress can change limits of this implicit power, however.
The Constitution is quite vague as to what exactly the President's powers are.
Perhaps. To the extent that it is true, it simply means that congress has that much more power to grant powers to the president or to refrain from doing so.