You can use XML.sendAndLoad() to store data on remote sites, if that site grants the appropriate permissions through crossdomain.xml.
This is how the configuration widgets store data onto the chumby site.
Download the example widget "SimpleBanner" and look at "simplebanner_config.fla", function "_chumby_set_widget_parameters".
Since this function sends XML as the content of a POST request, you'll need to process the data as raw content - it's not name/value pairs as POSTs are normally sent.
If you'd rather be more conventional, you can use LoadVars.sendAndLoad().