Posting for anyone who gets this error in the future.
Step One: Close Vortex in Task Manager (CTRL+Shift+ESC)
Step Two: Navigate to C:\Users\<yourUserhere>\AppData\Local\vortex-updater\pending.
Step Three: Inside C:\Users\<yourUserhere>\AppData\Local\vortex-updater\pending find vortex-setup-x.xx.x.exe and Right-Click, run as Administrator. In my case it was vortex-setup-1.12.3.exe.
Step Four: Go through the installation process, and then you're done.
