Today the Japanese government has released an app that can be used to show proof of Covid-19 vaccination. If you have the My Number card and remember your PIN, issuing the certificate is a pretty straightforward process. If you scan your passport, you can obtain a certificate for overseas use too.
In the App Store, the app mentions support for Apple Wallet, but there is no way to link your vaccination records to Apple Wallet from the app itself. As of the time of writing, you will need to manually download the QR code, send it to another device (or print it) and scan with your iPhone's camera app.
Hopefully adding the vaccination records to Apple Wallet would be made simpler in a future update.