Ecdesign does not require Admin rights to run but normally it's required when installing and updating.
It's stored in c:\Program files(x86) like most applications, many users are not allowed to install new applications but can alter them after the installation (like an update). But some users can't do that either, then you need to use admin settings to update.
The library (all objects) is stored on the user account, meaning if your admin account did the installation of Ecdesign, your own account won't have the library. The easiest way to solve this is by installing with admin rights (NOT as admin). Otherwise you could just copy the library folder to your own account. It's stored here: