This is most likely because your 'reaper-license.rk' file is located in reaper's resources directory in the following user path by default:
~/Library/Application Support/REAPER/
I don't think it's possible to have the reaper application in your Applications directory, and keep the resources directory in your local /Application Support/ path.
So, what you could do is make a portable install:
1) Log into your admin account
2) Make a folder called REAPER, in /Applications
3) In this folder, save a blank text file called 'reaper.ini' (must have a .ini extension)
4) Copy the reaper.app from the installer disc image to /Applications/REAPER
5) Put your licence file in this folder as well
6) Drag the icon to the Dock, and run it
This will auto-populate the new resources folder with all the reaper bits. But, the resources dir it looks for will exist in /Applications rather than a user path, so it should still work no matter which user account you are logged into