Is there any way to use Xcode on a computer that I don't have admin permissions for?
Is there a way to at least use command line tools like git and gcc?
There's an answer to this over on StackOverflow: https://stackoverflow.com/questions/1837889/authorize-a-non-admin-developer-in-xcode-mac-os
You need to add your OS X user name to the _developer group. See the posts in this thread* for more information. The following command should do the trick:
sudo dscl . append /Groups/_developer GroupMembership <username>
It looks like you'll still need the admin account for this part but after that you should be able to run Xcode on your account.
*This isn't the link from the original answer. That page is actually gone. I've added a new link ot a thread in the Apple Support Communities.
Hat tip to Ned Deily.