Is it possible to run unit tests in Visual Studio 2012 against a localDb that is built with Entity Framework 5.0 Code-First? Currently our development is against a SQL Express Server Instance but this will not work with AppHarbor deployments since the Db is unavailable.
I'm not sure that using LocalDB would solve your problem. To use LocalDB, LocalDB still has to be installed on the machine running your unit tests and that may not be the case on AppHarbor.
Consider these resources for alternatives: