This question is straight out of chapter 20 of the third edition of Matter & Interactions by Chabay and Sherwood (Wiley, 2010). The deep, underlying issue here is that for the first time ever that I'm aware of, an introductory physics textbook correctly teaches that a capacitor's fringe field must not be neglected! If it is neglected, then capacitors simply cannot work they way most introductory physics textbooks claim they work. Understanding the charging of the capacitor requires a thorough understanding of chapter 19 and the feedback mechanism described therein. The feedback mechanism drives the surface charge distribution and the circuit behaves accordingly. You're quite correct that the physical gap between the capacitor's plates creates a conundrum, at least at first. Then, you realize that there is an electric field between the place and, more importantly, a fringe field in the wire just outside the plates.
So, go back and make sure you thoroughly understand chapter 19 and then apply those ideas to chapter 20. The text does a quite excellent job of illustrating all this, so be sure to follow along, actually READ the text, do the exercises AS YOU COME TO THEM IN THE TEXT, complete the accompanying diagrams yourself without referring to the text. Here's a VERY important hint: you need neither the concept of resistance nor the concept of capacitance to answer your question. All you need is the concept of electric field and the concept of surface charge distribution.
M&I is the ONLY, yes ONLY, introductory textbook that gets capacitors right so you're actually being exposed to some groundbreaking foundational material here. Take advantage of it!