You probably could, using the TCP/IP VIs and a copy of the ftp protocol spec. NI has the Internet Toolkit, but that's really for transferring file to/from an ftp server, not to be an ftp server.
I guess more to the point, why would you write your own?
What are you trying to accomplish? Can it not be done utilizing a 3rd party ftp server?
It can, but it would be nice to keep everything self contained, plus there may be legal issues with bundling third party software in with our own.
Both windows and linux have FTP servers that are capable of being installed from their respective manufacturers, can't speak to a MAC. Although the LINUX one is more powerful. So that should not be an issue.
I am only thinking about how much time you spend writing an ftp server, especially if you need to support FTP and SFTP, plus all of the various functions that are part of the protocol stack... then debugging. Not going to be fun and not going to be done in a week or two.
I might pursue the licensing issues before I took to writing my own. Although I fully understand the keep it in the house thing.