After I upgraded from 1PW 6 to 1PW 7, I could no longer fill in passwords in Vivaldi and got the "I'll restart my browser" window, which indicated that communication with the 1PW browser extension did not work properly. In other browsers (e.g., Safari), 1PW still worked the way it should.
Restarting the browser and/or 1Password, reinstalling the browser extension, etc. did not help, but here is how I was able to fix the issue on my Macs:
~/Library/Application Support/Vivaldi/NativeMessagingHosts/and rename "2bua8c4s2c.com.agilebits.1password.json" to "2bua8c4s2c.com.agilebits.1password_org.json" (just to make sure it won't get overwritten in the next step, in case you want to revert these steps)
~/Library/Application Support/Google/Chrome/NativeMessagingHosts/and copy the file "2bua8c4s2c.com.agilebits.1password.json" to
If 1PW 7 is working fine with Vivaldi for you, you don't need to do anything. It's possible that I had these issues only because I did something similar (instead of waiting for an official fix) last summer after 1PW's switch to native messaging to get the browser extension working again with Vivaldi.
But if you are experiencing the same issues with 1PW 7 and Vivaldi as I did, the steps above will hopefully prevent you from having to downgrade to 1PW 6.
1Password Version: 7.0.1
Extension Version: 188.8.131.52
Vivaldi Version: 1.16.1183.3
OS Version: macOS 10.12.6
Sync Type: Dropbox