JR has been making improvements to its online booking platform and travelers on wheelchairs can now apply for tickets online.
For Nozomi/Hikari/Kodama trains
For JR East operated trains (for Kanazawa, Niigata, Tohoku region and Hokkaido)
Users can submit their request for a specific train and the train company will reply back. Travelers will still need to pick up the physical ticket before travel, since E-tickets are not available for Shinkansen. Also the websites are only in Japanese at the moment.