In the simplified model you're being presented, current can flow because there's no intervening resistance. Note that this is a good model to use in normal practice, because you can never

Here's a more detailed model. It's harder to use, and still doesn't fully model any real circuit -- because no model that we can write down is complicated enough to fully model any real circuit.

If you analyze this circuit, with the four resistors set to finite, non-zero resistances, then you'll see that $v_s(t)$, $v_m(t)$, and $v_c(t)$ are all different, because current is flowing. If you then find the limit as $R \to 0$, you'll see that the current is exactly 90 degrees out of phase with the voltage and there's no voltage differences to drive it -- but then, there's no resistance to current flow except for the capacitor's reactive impedance.

In the simplified model you're being presented, current can flow because there's no intervening resistance. Note that this is a good model to use in normal practice.

Here's a more detailed model. It's harder to use, and still doesn't fully model any real circuit -- because no model that we can write down is complicated enough to fully model any real circuit.

If you analyze this circuit, with the four resistors set to finite, non-zero resistances, then you'll see that $v_s(t)$, $v_m(t)$, and $v_c(t)$ are all different, because current is flowing. If you then find the limit as $R \to 0$, you'll see that the current is exactly 90 degrees out of phase with the voltage and there's no voltage differences to drive it -- but then, there's no resistance to current flow except for the capacitor's reactive impedance.