I remember it was possible for me to install skype with a non admin user on windows XP. Now the school upgraded the system to windows 7 and it tells me I dont have admins rights. I choose an install path to my network drive.
I believe your easiest option would be to run Skype from a USB thumb drive using Skype Portable. Saves installing it and also, your logs will be kept on the thumb drive, not on the system, which is a plus.