As the default is Mozilla Firefox, you can manage to get unity webapps working in Chromium by installing a plugin.
To install this plugin, go to Ubuntu Software Center. Search for and install the unity-chromium-extension package.
NOTE: The default version in Software Center is 2.4.6 for Ubuntu 13.04. I recommend you install the latest version (2.4.7 so far) which is available for downloading from below link. This version fixes no gmail icon on Unity Launcher for Chromium 27 & 28.
Unfortunately, new mail counts and indicator integration is still not working. Probably best to wait some more days for a fix.