Most fanfiction sites will have something called story alerts or something of the sort. These story alerts are usually notices in the form of an email sent to the email address you registered on the site with, informing you about updates for stories you specifically followed or otherwise requested alerts for. On fanfiction.net for example, there is
this button, on other fanfiction sites the mechanism probably works slightly different or is accessed in another way. It all ultimately ends up with the fanfiction site sending you an e-mail containing the story URL of the updated story, which FFDL then will scrape and try to update. What part of this process doesn't work for you, exactly?