Is every normal state on $A''$ a vector state?
A state on $A^{**}$ induces a state on $A$, hence a representation of $A$ and thus a vector state on $A$ in the universal representation.
The normal states are the $\sigma$-weak continuous states and $A$ is $\sigma$-weak-dense in $A^{**}$, so if the state was normal then it is determined by its action on $A$. But we have just seen that on $A$ it agrees with a vector state — hence the normal state agrees with a vector state on all of $A^{**}$.