This can be handled fairly easily. The backend that accepts the content from users would not track any data related from the transmitting party.

From your post I can't tell if you'd still expect there to be user accounts or not, but to even provide an extra layer of anonymity, avoiding user accounts all together would be a good path to head down.

There are BaaS (Backend as a service) solutions that allow mobile developers store data in the cloud very easily. Some that I would recommend taking a look at are Parse (www.parse.com) and Azure (http://azure.microsoft.com/).

All that being said, without knowing any of the other details of what your app is supposed to do, it would be hard to provide a cost estimate. But I'd guess it would be on the lower-side of apps that require a remote backend.

If you want to talk more about this in greater detail, don't hesitate to set up a call.