Negative Charge Pump with Ring Oscillator
Have a look at this charge pump circuit using a 555 that produces a negative voltage:

http://www.eleccircuit.com/wp-content/uploads/2007/09/nagative-voltage-generation-using-555-timer.gif

In principle, the square wave oscillator doesn't have to be a 555 timer, it could be a ring oscillator using inverters as you propose.  In practice, using 4000 series CMOS logic for example, the negative output voltage produced would not be able to provide a great deal of current due to the current sourcing limitations of the IC.  If the amount of current required from the negative output created is small, maybe half a mA depending upon the supply voltage, I don't see why using CMOS inverters for the oscillator wouldn't work.