Mountain Lion - Does it allow different users to share database files using iCloud?
iOS and OS X only support a single account to access iCloud. It can't be done on a per-app basis.

Furthermore, the iCloud syncing stuff isn't really designed for concurrent multi-user access, it's built for the sort of syncing that needs to happen for personal devices - i.e. that you're using only one device at a time. It's also worth noting that from a developer's standpoint, iCloud is far from perfect, particularly with regards to the Core Data (database schema) syncing and storage.

You don't say exactly what your needs are, but I strongly suspect you'd be better served looking somewhere else for a business-critical app.