Is it even possible to run a VirtualBox instance on a simple shared web hosting account without root access?
I’d love to see that!
It's possible to run VB in headless mode inside a VM (albeit w/o CPU acceleration) but not in a "shared" web hosting subscription, which is essentially a non privileged *nix login to a host.
VB does require 'root' access for its kernel modules, (output of lsmod below) plus building them via DKMS for kernel updates.