Yes, it's definitely possible. You could also share them over the web directly via the NAS. To do it from the LAMP system, you just need to mount the filesystems on the LAMP machine (likely via NFS) and configure your webserver (ftp, ajaxplorer, etc) to use those mounted directories to serve files. This would basically be the same approach as if you wanted to serve files directly from the LAMP machine.
This is a fairly common approach, and for a home setup there aren't really any caveats, it should just work.