Not exactly, it will only provide one half of what you need. There are a variety of ways to do this, two 12V Wall adapters will work (one to provide +12V and the other to provide -12V) it needs 2 power sockets (which may not be an issue). That may be the easiest and it should work. One of the adapters provides +12V and the other has its +12V lead attached to ground and its ground lead connects to the -12V lead on your breadboard (providing -12V relative to ground). Another option is a regulated 12V wall adapter and a boost regulator module that will take 12V input and produce a 24V regulated output. Ground for the boost module connects to +12V and the 24V output provide the -12V output. This unit from pololu will do it but is quite
expensive at $11US,
an similar unit from ebay is $1 US (but long delivery) such as this one:
In either case you would need to adjust the supply output to be 24 V (as both boost modules are variable voltage). A 5V wall adapter and one of the ebay modules I referenced earlier will work as well with no voltage adjustment required (as the module takes 2.5 to 5V in and outputs +-12V already), the 5V adapter (or 2 AA batteries at 3V) will power the module and it will produce the needed + and - 12V to power your circuit. The downside to it is shipping from China for the module can take up to a month or so via mail, so it isn't necessarily fast if you are in a hurry and as the module is only $2US overnight delivery would be probably expensive. You could also make a split power supply from a 24 volt center tapped transformer, rectifier, capacitors and 7812/7912 regulators (there are instructions on google for doing so), but you don't seem to be able to buy a cheap +/-12V wall wart from any supplier I can find. You used to be able to find them surplus sometimes, but even my favorite surplus suppliers don't have any.
Peter