Hello Becky Carr,
You can edit the attribute table in a web map as long as the web map contains an editable feature layer. It is possible to edit the feature layers directly through the item details page, data tab.
It is also possible to edit existing attributes through the edit widget. Unfortunately the current version of Web AppBuilder deployed with portal does not support editing attributes using the Attribute table widget. There is an existing ideas post which I've linked below.
Hope this helps!