Access to the feature is more subtle now. As described in
- Launch Health.app
- Tap 'Summary'
- Tap your profile picture
- Tap 'Medical ID'
- Tap 'Edit'
https://support.apple.com/en-us/HT207021Source, you launch health.app, hit 'summary', then, unexpectedly, hit your profile picture. Now hit 'medical ID'. Then hit 'edit'.