I am not 100% sure about version 7, but I believe the labels should be in the Extensions/Labels folder, right next to the application. Perhaps you could simply copy them over from the other system.
Thanks for the insight on the location of the file. That was helpful. I ended fixing this problem, but not from copying the file.
I was using FileMaker as a regular user without admin rights. So, when I tried to copy the file, access was denied. Then I logged in as admin, and starting thinking about permissions. I looked over all the permissions in Program Files -> FileMaker -> FileMaker 7 -> Extensions -> Labels and decided all was well (users are able to read files, but not write). Out of curiousity, I opened FileMaker as admin and found the labels were loaded. Then, without having changed any files or permissions, I logged back in as a normal user and the labels appeared in a list there, too.
I double checked myself on another computer with the same problem. Log in as admin, start a new labels layout in a file, logout, and works perfectly for the non-admin user. Something in the background must be getting triggered when I start the new labels layout. If anyone can point out why this makes things work, I'd be interested just as a system admin who likes to know how his systems tick.