Not sure if you can export shapefiles.shp from Business Analyst Online. It may depend on product's package subscription. If there are not many attributes to tag, then you may be able to georeference a screen shot and create shapefiles. Time consuming yes but affordable too.
It is possible to ultimately get a zipped shapefile of a BAO study area but it is has many steps and is not the most intuitive workflow. Basically use the share map to ArcGIS Online. Then go to my content at AGOL and you will find the web map and open it up in a new map. Then you will need to save as a layer to create a feature layer. Open this in a new map and then perform an analysis>manage data>extract data. Finally one will be able to download a zipped shapefile from My Content.
I find it best to understand that workflow so you can extract data, or create layers from data, in BAO. Then save to ArcGIS Online for more advanced styling of your map, analysis, and preparing it to share with the intended audience. From there you can choose to export, print, share as web app, etc.
BAO isn't meant to be a complete solution, but the app is a tool to extract data, reports, & maps you need for further use. It's one of many add-ons to the ArcGIS mapping system that you can take advantage of.
