Bridged will bypass the host OS's network stack completely, if I understand correctly. See
Virtualbox Networks: In Pictures: Bridged AdapterVirtualbox can be installed without Bridged networking. If your IT dept has sufficient control of what and how programs get installed on the computers, it can turn off Bridged during Virtualbox installation or upgrade, then no VMs can Bridge.
I understand it is also possible to configure a fancy-enough network switch to not allow more than one computer to network through the switch port, which would also block Bridged, since Bridged VMs would appear as separate computers to the network. This setup is controlled in the switch's configuration, not by Virtualbox settings or installation, and might be even more controllable than setting Virtualbox installation parameters.
NAT and NAT Network do go through the host PC's network stack, so they should comply with host networking restrictions.