Recently Oracle has released the new version of VirtualBox and I installed it on Bigsur. Finally, I could run VirtualBox on BigSur.
The following are what I got when I tried to install Ubuntu on VirtualBox.
In my case, the issue was the security settings. As you know, macOS gets more and more strict.
Here are what I did to work out the issue I had.
Step 1 Run the command
$ sudo kmutil load -b org.virtualbox.kext.VBoxDrv
Step 2 Allow VirtualBox/Oracle to have permissions
System Preferences > Security & Privacy
Unlock and Click Details and checked Oracle
Privacy tab > Full Disk Access
Step 3 Reboot macOS
Just restart macOS and run VirtualBox