Short answer: You can't. (Sensibly, anyway.)
JR is not a single company, but whole bunch of them, and their integration is less than stellar. The Tokaido Shinkansen between Tokyo and Kyoto is operated by JR Central (Tokai). While they do offer an online booking service, it's -- rather incredibly -- a paid, members-only service that requires registration and a yearly fee (500 to 1000 yen), and it's Japanese-only; probably more hassle than its worth.
The second option would be the all-Japan Ekinet service, but this too is Japanese only and requires signing up, and its (bizarre) Achilles heel is that you can only pick up your tickets at JR East stations like Tokyo. Kyoto is in the JR Central zone, so you're out of luck. Also, there used to be an English version of this, but it's been shut down. (Why? I have no idea.)
Third option, and this is going to be a long shot, is to contact Lotte JTB, the Korean branch of Japan's largest travel agency JTB. They offer an English service, but they appear to focus almost entirely on selling stuff in Korea, so I doubt they'll be able to reserve a seat for you. But give 'em ring, it won't hurt.
The one silver lining to this is that even during Golden Week, the trains don't fill up entirely, you should be able to find a seat even at short notice if you're flexible with your travel times and/or willing to upgrade to Green Car. And if not, you can always get an unreserved ticket and stand -- not particularly pleasant, but it's only a bit over two hours.