Hi, thanks for your interest.
The mapping.xml used for SubmitToSharePointList contains the SharePoint List GUID (for the list you are submitting data to). This GUID will be different for a new site, even if the name and columns remain the same.
You could:
1. Update the GUID in the mapping.xml before deploying the form to a different site.
2. Add form logic to the form to automatically detect where the form is being opened from and which list GUID should be used in the mapping.xml.
If you don't have a lot of sites, the first option is easiest. If you have a LOT of sites, then #2 is better.
Good luck!