A capacitor without a dielectric is connected to a battery at t = 0...

When a capacitor is connected to a battery, conduction current flows through the connecting wires during the charging process. Since point $A$ lies in the connecting wires, conduction current flows through $A$ as long as the charging is incomplete.

Between the plates of the capacitor, there is no physical medium, so conduction current is zero. However, as the capacitor charges, the electric field between the plates changes with time. This changing electric field produces a displacement current given by $I_d = \varepsilon_0 \dfrac{d\Phi_E}{dt}$. Since point $B$ lies between the plates, displacement current flows through $B$ as long as the charging is incomplete.

Therefore, current flows through both $A$ and $B$ until the capacitor is fully charged.