I'm building a 5V power supply and I'm hoping to be able to use the 9 VDC, 1 A wall wart that I have lying around (it actually puts out 12VDC). I'm using the following circuit and getting a good response on LTSpice, but I'm wondering if I need to add a bridge rectifier at the front end or if that is only necessary when using a transformer?
You don't need a bridge rectifier unless you want your circuit to work with either polarity on the input.
Depending on how much output current you need you may want to consider a switching regulator rather than a linear one. With 4V or more across your linear regulator and 500mA out you will be dissipating at least 2W in the regulator. If you need the high PSRR and low noise of this LDO, you could pre-regulate with a switcher so you're not dropping as much voltage across it.
The 2W dissipation is do-able, if you manage PCB layout and thermals correctly.