To clarify what Jake is saying for anyone else who finds themselves here: ArcGIS Online accounts assigned the 'Data Editor' role cannot export layers, because they cannot create new items. Administrators must upgrade users to the 'Publisher' role in order for them to be able to see the Export Files button.
* And, yes, this is not intuitive, since the Data Editor role is listed below Administrator and above Publisher in the Roles drop down. Upon closer inspection, this appears to be an alphabetical ordering, and not the hierarchy it implies.