I have made a Full install bootable USB flash drive.
I would like to use it to install Ubuntu to a friends computer.
Can I install the Ubiquity installer to this flash drive?
You can install Ubiquity and that in turn will install all the lower level packages it needs to install Ubuntu to another drive.
sudo apt-get install ubiquity ubiquity-frontend-gtk
Running it is a manner of running
ubiquity --desktop gtk_ui
The manpage is a bit inconsistent on this, but it looks like you don't run it as root, and it will call gksudo itself when necessary, but if that doesn't work try calling it with gksudo.