If they are going to be updating announcements on a regular basis, seriously consider a CMS alternative like a Wordpress.
While it is possible if you make the file publicly available and put the URL as an include (ie: in PHP - include([urltodropboxfile]); ) it really isn't recommended. First, text in a txt file needs to be parsed by the server, otherwise it will come through as one big text block. Because of that it would be advantageous to just have your client make HTML files. And second, most shared hosts, which I would assume is what your client is using, disallow remote URLs in includes for security reasons as well as bandwidth reasons because they don't know what the process being included is. To test this create a php info file and check for the variable "allow_url_include". If that option is not On or allowed, then you can't include Dropbox files even if you wanted to.