Determine every vector field such that its field lines are contour lines to $g(x, y) = x^2 + 4y^2$ Is it possible to determine every vector field such that it's field lines are contour lines to $g(x, y) = x^2 + 4y^2$? If so, how?

At each point ${\bf z}=(x,y)$ the gradient $\nabla g(x,y)=(2x,8y)$ is orthogonal to the contour line of $g$ through ${\bf z}$. Turn this gradient by ${\pi\over2}$ counterclockwise to obtain the vector field
$${\bf v_*}(x,y)=(-8y,2x)\ .$$
We can still multiply this ${\bf v}_*$ by an arbitrary nonzero function $\rho(x,y)$. In this way the most general answer to your question becomes
$${\bf v}(x,y)=\rho(x,y)(-8y,2x)\ .$$
Of course ${\bf v}({\bf 0})={\bf 0}$ since there is no actual contour line going through ${\bf 0}$.
