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Virtualbox is an open source application and it would be amazing if you could make it portable!
Thank you for your time.
however bear in mind, that virtual box only useful when all needed drivers are installed on the host system. So definitely you need admin rights on the host system.
from the website:
VirtualBox needs several kernel drivers installed and needs to start several services: if the drivers and services are not already installed you’ll need administrator rights to run Portable-VirtualBox.