I am not a writer, but I have developed a website for them. Where can I advertise and publicize it?
Why, wherever writers can be found. :)
Various writers' forums will have places for announcements. You could try approaching writing blogs - emailing blogger, asking for a link to your site if they find it useful. There's magazines like Writer's Digest where you can advertise. Here on Writers.SE, the chat room is the most appropriate place.
But really, beyond such rudimentary guidelines, you'll need to actually do some market research. I'm somewhat dubious about a site "developed for writers" if the development team doesn't know enough about writing to find some popular focal points. Find the places where people are talking about the problem your site solves, and advertise there. No?