As @mark pointed out, standard users cannot set up Time Machine and do many things needed to maintain a system. Therefore, all a standard user can do is copy files or install an application in the user folder to set up backups of the data that is readable by that user.
I have only tested CrashPlan as an administrator, but I see no reason why you couldn't use that to get incremental backups over time of one user folder (or a subset of that folder). You can install applications in your folder as a normal non-admin user.
By making a standard user - the admin is responsible for letting you do anything that is in your scope and in this case - Time Machine was allowed for you by virtue of making your account, so that's why you get to start that process.
The fact that you have access to the backup media has nothing to do with software or permissions. If you choose to take that data, you can simply do that without any concern for how the system is set up or designed.