Hi Ethan,
ArcGIS Explorer cannot create or edit shapefiles I'm afraid, and this will not change in the forthcoming new release.
If you have ArcGIS Desktop available, that would be the best way to create a shapefile. Using the Explorer SDK it would be fairly simple to write an add-in that exports a folder of Point notes to a CSV text file, which could be imported into ArcMap as an X, Y layer - at that point you could use ArcGIS Desktop to perform the conversion to shapefile. I imagine there are also geoprocessing tools to do something similar (although I'm not very familiar with the range of tools available).
