Hey taarna,
You should simply be able to install new vegas on your newer machine. Use the steam account you originally activated the code on.
You can install steam on as many computer as you want. Just sign into the original account on that computer. DON'T make a new steam account.
You can install fallout new vegas on multiple computers. You probably will get a prompt indicating you are runing new vegas on a new machine.
They will email you a small code and you'll have to copy and paste it into the textbox steam gives you. You should be good to go after that.