What you can do is having a 5V regulator with its input on the 9V, use this 5V output for some circuitry such relays, and using this 5V as the input of a 3.3V regulator where its output is for the ESP itself.
Damian Allen wrote:That's actually what I figured. The problem is, all the other circuitry is going to be intermittently driven (PWM). How well would a linear regulator handle rapid voltage changes (say 240 pulses per second)?
Are you sure you want to use a linear regulator, instead of a switching regulator?
In any case, adding a capacitor in parallel with the regulator's input should solve the issue.