Explain how would the energy conservation law fail if electric current is assumed as a vector quantity?
Perhaps you need to double check the question? Or provide greater context?
The fact of the matter is that electric current IS a vector quantity, and the energy conservation law does NOT fail. So some added detail is needed to answer your question.
Current is a vector quantity because it is made up of moving charges; the charges are moving in some direction; direction + quantity = vector quantity.
Consider the energy conservation law written in the form of Poynting's theorem:
du/dt = -∇·S - J · E ( · is the dot product )
Here, J is the current density vector; note that I, the current vector, is qJ. So I, even as a vector, satisfies the conservation of energy. Note that current is always a vector, even if it is all flowing in only one direction.