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.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/1p6Nl.png