NASLite is only visible to machines on the LOCAL network. It doesn't support a gateway in version 1, which means it doesn't know to use your VPN router to connect with other sites. The other stores shouldn't be able to see the NASLite machine, so they won't be able to read/write/delete anything.
On the other hand, any computer at your local store that's on the same local network as the NASLite box will be able to see the NASLite machine. I would guess you only have a handful of computers at your local store. Only people authorized to use those computers could do anything to the NASLite box.
You didn't mention which operating system you're using, so I'll guess it's some verison of Windows. If you only configure one computer with a mapped network drive access, the people using the other computers wouldn't have your NASLite drive show up unless they normally use the "view workgroup computers" command.
For backups, you would have to get the files from your remote stores onto a local computer, and then move the file from the local computer to the NASLite storage. If you're on a 10Mbit ethernet network it could take a while to back up 20GB. The free floppy version doesn't do DMA so transfers are about 200kB/sec versus 9-10MB/sec with the CD versions.
If you're just trying to get a floppy already made, there is a guy selling pre-made NASLite floppy disks on E-bay. Or as was previously mentioned, you can buy the CD that has a program to make the floppy.
If you just need to make a backup copy of your backups

you might look into external USB or firewire disks. You could physically disconnect them and turn them off except when you want to update files. The downside is they are much easier to "borrow" from the store

unless you have some way of securing them.