You can of course write a folk song. It's a style or set of styles, nothing more, nothing less.

Jim is correct however when he mentions a way of it becoming a standard, i.e. For people to sing it assuming it has always been around. Ewan MacColl was a master of the art. Try telling people his songs aren't folk and you can have hours of fun being laughed at. John Connolly reckons he hovers between pride and frustration when his Fiddlers Green is assumed to be traditional (or as per many lyrics websites, even ruddy Irish.)

Watch out though. The price of writing standards is that everyone assumes you are dead. I have lost track of the times I have heard people sing The Final Trawl and tell the audience Archie Fisher is no longer with us. (True, I think he is touring abroad at present.)