If you simply swap the hard drives you could run into a number of problems:
1) Vista is subject to Microsoft's highly irritating hardware detection system intended to stop anyone using the operating system on more than one computer (even if, like you, one is not intending to do that ). You will have to join a queue of other poor souls trying to get permission to use their computer from some undermanned call centre.
2) More properly, the operating system may have difficulty adapting to the different hardware on the new machine because Windows was set up on older hardware.
It is possible that it'll work but it may not work perfectly.
Vista is anyway, not the best version of Windows -- XP was better and Win7 is better than that. As Win7's extortionate retail price will be somewhat concealed or reduced if you buy the computer with it on already, I would advise doing that.
Of course you might want to have a look at Linux as a now workable alternative to Windows -- probably Ubuntu is the safest bet and simple to download for free.