I'm sure someone will have a better answer than this, but a 'sideways' answer is that Filemaker can import from an Excel file, automatically, silently, so you could at worst maintain an Excel worksheet that simply does the data retrieval from the web as you currently do, then Filemaker imports from it directly, to a schedule, if you want.
If nothing else you could use this technique to develop the downstream part of your Filemaker solution while you wait on the 'proper' answer.
I know it is probably not what you want, but you can view the web-page live in a Filemaker record, too.
Thanks much! This is a good solution for now. Maybe someone will know of a plug-in.