Exact Audio Copy, like some similar programs (I know older versions of Nero are like this) can be made to work on a Windows limited user account, but it requires a couple of things.
First, you have to make the DAT files in the EAC folder writable by all users (or at least those you want to be able to use it).
Second, you have to enable low-level access to the disk drive for limited users. The easiest and least dangerous (in terms of downloading software from who knows where) method of doing this is to simply use Nero BurnRights (search down on the page), and it will allow you to set this option; note that you don't need to actually have Nero to use this program. Of course you'll have to install it and run it as admin, but once you set the option you'll be able to use EAC (and other similar programs) from any user.
Finally, make sure EAC is set to use the Native Win32 Interface (EAC Options->Interface); this should be the default, but who knows.