As Max Erickson says, they need their own OSM account. You can write your own editor, which submits changes to the OSM API. Users can authorise your editor to edit in their name via OAuth. Once someone has auth'ed your app, they can stay entirely within your app.

But be careful that you don't lead your editors to think this isn't a global database that others use, or other OSMers will be annoyed.,

No. The person doing the edits should be directly contactable through the OSM messaging system.

You can act as proxy like WheelMap does (upload the edits on behalf of the users), but you must then be reachable when some edit is wrong.