Can I install Lightning even if I don’t have administrator privileges on my computer?
Yes, you can.
This is because Lightning is built upon a safe technology designed by Microsoft: it’s not independent software, it’s an “Office personalization”. Personalizations come with security limitations. Windows allows you to install them even if you’re not an administrator.

On top of using a secure software format, Lightning does not collect personal data. When telemetry is activated, Lightning collects usage data (e.g., what function or keyboard shortcut you use). Lightning does not collect any type of confidential data such as in-document text, file names… You can turn off telemetry by going to File > Add-ins (on the bottom left corner) > uncheck “Enable telemetry”.