I was messing around with the arcgis.features.analysis.join_features function that Jayanta mentioned, but whenever I used it, it would not figure out which polygons the point data was in. It did join all the fields, but that's not what I was looking for. I may have been going down the wrong path.
This will end up being a once-a-day automated process, so I would prefer to do it in Python.
What I ended up doing was use the arcgis.geometry.Geometry.contains function to determine if the point was in the polygon. My data set will always be small, so it's not a performance issue to loop through my 7 polygons and 4 or 5 points to figure this out.
I appreciate the suggestions!