Hi Michael,
Yes, qRules can submit to lists on different sites. When creating the mapping, make sure to select the option to use the list name. Of course, make sure that your lists (on the different sites) have the same name and the same columns so that the mapping to one list is compatible with all the lists.
Then, on your form, before executing Command = SubmitToSharePointList /mapping=mapping /submit=Submit, make sure to set the URL to your the site you're submitting to, in the mapping secondary data source: /myFields/SharePointListURL
Hope this helps! There's a free trial of qRules available on Qdabra.com.