There is the Peer Production License, but the only project I’m aware of that is using it is looking at using another license.
My suggestion would be the AGPL:
The GNU Affero General Public License is a modified version of the ordinary GNU GPL version 3. It has one added requirement: if you run a modified program on a server and let other users communicate with it there, your server must also allow them to download the source code corresponding to the modified version running there.
It wouldn’t stop non-coop private competitors from using the code but it would force them to share any changes they made to it, so if they added some additional code for some new feature you would then have access to this to incorporate it into the core codebase — sharing would be enforced, I think this is probably as good as it gets in terms of co-operative software licenses.