This website provides information about clinics / hospitals approved by government of Japan for Vaccination
You can search by your prefecture > City
You can visit the Hospital / Clinic website and call to their phone number for making a booking or search the address in google and get their phone number and call.
I personally called many clinics in Edogawa area and most are booked full for July/August and some have availability in September.
You don't need to have Vaccination coupon for making a reservation. Most of the clinics will ask to present Vaccination coupon when you go for the vaccination on the scheduled day.