How to power a circuit that needs +9VDC, GND, -9VDC?
You will probably find that two 9V wall warts will do the trick - connect the positive output from one to the negative output from another - this connection is the "new" 0V and you'll have +9V and -9V available - regard it as two 9V batteries in series with the middle (common) connection being called 0V.
Because wall warts produce isolated output voltages, they can used like this but always double check with a meter first to ensure everything looks and feels OK.
Looking at your op-amp circuit you should be able to get aways with a single 12V wall wart and a rail splitter made by two resistors across the supply. The junction of those resistors will be 6V up from the negative rail and ALL the points marked with an "earth" symbol on your circuit can connect to this point. This is a much used technique but, as always, it won't suit every target circuit but looking at your circuit it should be OK.
If you have a transformer-based PSU available you can probably convert it from single to split-rail as shown below.
simulate this circuit – Schematic created using CircuitLab
Note that each rail is now only half-wave rectified so I'd recommend some large caps or voltage regulators to eliminate hum.
If you want to keep a standard jack on the PSU then convert it to an AC PSU by removing the rectifier and capacitors and put the diodes and caps into your project case.
This configuration can also be used to make an 18 V PSU.
simulate this circuit
If you're wanting to produce a split-rail from a single supply, you might find DC-DC inverter ICs useful. Such as these from Microchip:
http://uk.farnell.com/microchip/tc7660hcpa/dc-dc-volt-converter-7660-dip8/dp/9762698
I tried these DC/DC converters in an audio project with single supply limitation (guitar effects pedal) and they worked a treat. Here's an example of a simple audio project that uses the 7660, hopefully it will be of some use to you!
http://www.generalguitargadgets.com/effects-projects/distortion/kc/