The project got created and would build with no issues, but when I tried to run it I got the following error.
Call to Node module failed with error: Error: Cannot find module ‘./wwwroot/dist/vendor-manifest.json’
webpack --config webpack.config.vendor.js
After running that command everything worked like a charm! It is worth noting that the command above can be found in project.json in the scripts prepublish section. From the little bit of reading I did it looks like this command should be rerun anytime a new vendor dependency is added.