In the future, I would like to be able to make changes, updates, etc. to the map the contractor created for us. Is it possible to transfer the map from his account to mine in ArcOnline?
Is there more than one way to do this? What is the easiest way?
In effect, you have to change ownership of the web map from the contractor to you. When you look at the map item details page, on the right hand side, down the page you should see Owner and the option to Change Owner.
Select Change Owner and select yourself. This will change the ownership of the web map. You will need to change the ownership of the underlying feature layers aswell.
Your ArcGIS Online membership must have the Administrator role or a customised administrator role ( with Administrative Privledges > Content > ReAssign Ownership enabled)
Have a look at Manage content—ArcGIS Online Help | Documentation, Change Owner section for more information.
I assume that you have one ArcGIS Online site and the content lives in here, we aren't moving from one ArcGIS Online to a second ArcGIS Online. Use Distributed collaboration—ArcGIS Online Help | Documentation if you wish to move\synchronise data between sites.