The Oracle client software is not installed on AppHarbor, but you can add it as a feature request here. It is however rare that application specific software is installed on the application servers, so figuring out an alternative solution is probably the way to go. A couple of suggestions:
Bin deploying:
According to this discussion the client library needs to be installed, but you might want to test the approach "Fat Deployment" approach described in this thread.
Managed ODP.NET library:
It seems like a fully managed ODP.NET library from Oracle (that won't require client software to be installed on the servers) is in the works and there should be a public beta available sometime soon. There are more details on the Oracle forums where you can also contact the ODP.NET product manager to be included in the private beta.