Can/should we get rid of ArcMap by transferring all data and maps to the ArcGIS Online format? With the intention of reducing costs, would this help reduce costs or is it the same cost to have both versions? Thanks.
With regard to cost I would discuss this with your Account Manager or our Sales team.
There are certain workflows that require Desktop (like running models or advanced geoprocessing tools, creating advanced layouts, etc.) and others that require ArcGIS Online (field data collection, sharing Web Apps, etc.). You can store data in both Desktop and ArcGIS Online.
Looking at some of the overarching documentation might help with the decision: What is ArcGIS Online?—ArcGIS Online Help | Documentation, Desktop: ArcGIS Desktop | Desktop GIS Software Suite - Esri
Hope that helps,