Can I use an op-amp to supply itself a negative voltage?
You can generate a negative voltage with a small current quite simply with a source of AC, and a voltage doubler rectifier.

(This shows a doubler, reverse the diodes and capacitors for an inverter)


See here for some circuit diagrams.

The AC would come from your Arduino, by toggling a single pin regularly. You must add a series resistor too. It might be possible to use the PWM outputs, set them for 50% (AnalogWrite) and the pin will change state continuously. 

Notes: as you are producing a square wave, not a pure sine wave, you will need a resistor to limit the current, as though the uC was driving a short circuit. If you only have 5 V, you should use Schottky diodes to get the most possible voltage out. Choose appropriate capacitor values for the frequency of AC that you have. Higher frequency is better, until you get to the point that the junction capacitance of the diodes becomes relevant. Available current will be small, perhaps 1/2 or 1/4 of the maximum output current of the Arduino, be careful not to overload the output pin, even during start-up when the C2 output capacitor is not charged.