@Gwen-Dragon Thanks for the reply! I did some digging and found a pertinent blog post that I was able to use to get it working (and another source of useful info). In the Registry Editor (so use this information solely at your own risk), I:
Removed HKLM\Software\Microsoft\Windows\CurrentVersion\Explorer\AppKey\7
Added HKCU\Software\Microsoft\Windows\CurrentVersion\Explorer\AppKey\7\ShellExecute = the path to Vivaldi.
Works fine now! The REG file is below in case it helps anyone.
Windows Registry Editor Version 5.00
[HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\AppKey\7]
"ShellExecute"="C:\\Users\\me\\Vivaldi\\Application\\vivaldi.exe"
Save it as WebHomeKeyVivaldi.reg, edit it in Notepad to change C:\\Users\\me\\Vivaldi to wherever your Vivaldi installation is, and double-click to merge it.