Yes you can, but there are pitfalls. Most of which involve microsoft.
To move an existing OS, you just move the drive, do a repair install, and install the new motherboard drivers. On vista, you just do an upgrade install over the existing OS since the repair install function on NT 5(2000/xp/2003) was removed in NT 6(vista/7/2008). Thats the easy part.
The problem is that the vista OEM license is non-transferable. It would ask you to reactivate windows, which with those keys could only be done over the phone. In all likelyhood they would not do so since the OEM license cannot change motherboards.
However you said something about windows 7. Now you CANNOT upgrade from 32 bit to 64 bit on any version, but you can upgrade from vista 32 bit to Windows 7 32 bit. Tell me more about your copy of windows 7 and your plans for it.
By the way, what happens to the old machine after you move the drive?