There is no way to do it online for non-Japanese speakers.
You can call as noted above (I've never tried that), or just head to the restaurant first thing in the morning. Things like the Tahitian Terrace and Diamond Horseshoe are usually already fully booked in advance.
Other restaurants, like the Queen of Hearts Banquet Hall, Blue Bayou, and Magellan's, you can get into with no problem if you eat lunch early (10:30 am--not so odd if you were on line at the front gate at 8:15 am) and dinner at 4:30 to 5.
We have used skyped to call the resort. You have to call one number for park restaurants and one for the hotel ones. But honestly if you go at off times they take you regardless.
Bringing the Love since January 10, 2011
We were sent over there with the message that only we would defend Disney quality.. We learnt quickly that the Japanese culture speaks to a level of quality that is hard to comprehend. They were all unbelievably skilled and willing to do the right thing.
Craig Russell -Walt Disney Imagineering