Chris,
The best way to do this would be to write an MapInfo Pro addin application to present and UI and call the API to get the resulting polygons. As Peter said, using .net would be easier to talk to the web service. After that point you would use MapBasic to add the polygons to a map for display and interaction.
We have done the same thing with our Pitney Bowes LI Api geocoding service. We have an sample application that shows how to call other LI Apis.
[Forward looking statements follow]
We are now working on doing the same with our GeoZones routing Api for our next release. In addition we plan to expose a .net API for creating/editing MapInfo tables, even from a background task. This would make it easy to get the resulting polygons onto the map.
?