I found the answer to this, in case someone comes along with the same issue. This is my trial and error interpretation, but it does work although I cannot find it documented anywhere.

If you are using a Flex Card that calls an Integration Procedure (IP) that in turn calls a Load Data Mapper (DM) you need to have a Data Source, even if it is trivial. I used a Custom JSON that holds one empty field.

If you don't have a data source the card will not put the data from the input fields in to the JSON structure it passes to the IP. It does not matter if you set input mapping it still will not work. This is only an issue of the combination of an IP and Load DM from the Flex Card. You can call the DM directly without issue if you can get by without the DM.