After a bit of searching I found a pastebin with the answer, to load the kernel driver run:
sudo kextload /Library/Extensions/VBoxDrv.kext/
Attempting to start the virtual machine progressed a little further, but I was now greeted with:
Clicking OK allowed me enter the settings, but I couldn't change anything. The solution for me was to discard the saved state for the VM, go into the settings and disable the host-only adapter (not sure if this step was necessary), then boot the VM.