The input voltage will be 12V from a PC PSU, will the LM317 be able to give 12V output?
No.
You need a voltage greater than 12V say 14V to get 12V out. Also you have no input or output capacitors and have you allowed for the Vssat voltage between collector and emitter when the transistor is on?
You also need resistors in the base of the transistors to limit current flow.