Yes it can be done. There are various options depending on the fault. I have got around every fault the ignition module can give someway or another.
If the key turns and but the steering lock doesn't unlock you can buy a emulator for this and wire it to ignition live or have it on a switch.
If the key turns, the ignition turns on and the starter turns over this is probably the easiest to overcome, the immobiliser can be removed from the engine ECU so the car will start all of the time.
If the key doesn't turn and everything is dead this is obviously the hardest to overcome but it is certainly doable. You need to remove the immobiliser from the engine ECU, feed each required ECU/module with an ignition live and sometimes you may need a CAN bus module/emulator to transmit the correct messages to let the other modules and ECUs know when to turn on and off.
Obviously a better solution would be to either get your unit repaired at the fraction of the cost of a new one, getting it repaired has the benefit of not needing programming so you can fit it yourself.
You could also fit a second hand unit, you would need to get it programmed and I'm not sure a main agent can do this but any good auto-electrician would be able to.