The Japanese government website for vaccination locations and reservations.
Online Vaccinations Via COVID-19 Vaccination Navi
Online reservations are available through COVID-19 Vaccine Navi. Their website isn't exhaustive, but allows you to make online reservations with government-sponsored sites. If navigating the page in English:
Select "Search Vaccination Sites" -> Your Prefecture -> Your Area
Scroll down for the list of vaccination locations. (Addresses and names will likely not be in English.)
Next to the location you'd like to reserve with, select the + button to open the drop-down menu for that location's reservation options. Some only allow reservations by phone, rather than online.
Click "to Reservation page" if the location is compatible with online reservations.
You will need your voucher number in hand to complete the reservation.