It looks like this is the way it will always be, at least for the foreseeable future. The New York City Department of Finance does not seem to allow e-filing of the NYC-204 return. It has to be mailed to the Department of Finance.
There might be a way to file it online on the NYC Department of Finance Business Tax e-Services web site. But that would mean entering everything on the web site. There is no way to e-file the NYC-204 that is prepared in TurboTax Business.