Joe Tomei here. If anyone hasn't gotten their jab, the prefecture has opened a vaccination center (Moderna) at Gran Messe (near the airport expressway entrance) for anyone who has a coupon
This is the reservation link
https://www.covid19-vaccine.mrso.jp/kumamoto/VisitNumbers/visitnoAuth/
The hours are weekdays 6pm-9pm and weekends noon- 6.
Mike Rupp, in the comments, writes about the Vaccine Passport
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You go to the Kuyakusho, and they can give it to you at the window where they do Japanese passports. You need to bring your passport, the proof of 2 shots (the sheet with stickers) and you must write down a date for which you plan to travel on the form. It took about 15 minutes for me [Mike] and I was the first foreigner.
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Mike's timeline has a photo of his document but unfortunately, I can't add it to this post ($%&'&$ Facebook) and we've blocked photo comments ($'%&#%& trolls on the Internet), but if you ask him nicely, he may help. If you aren't sure of your Japanese (or even if you are!) having a photo of the object of your desire can really help things along.