There are two types of OEM activation. The first works like a full retail key, but only once; this is the key that comes on the COA sticker, and is a single computer activation which activates online with microsoft, and ties that key to the hardware fingerprint of that computer.
The second is a PAK (pre activated key); the OEM makes a custom disc with their PAK appropriate to that version, and a digital certificate specific to that OEM, and it checks it has a flag in the bios that matches the digital certificate. If any of the three are missing, it won't automatically activate.
So if you get an OEM disc as part of the upgrade offer, you won't be able to use it on the 2nd computer with the PAK built in, as your BIOS won't match. However, you should still be able to use the OEM key that comes with it on the COA sticker on your 2nd computer. As long as you only use it one of the two computers, you'll be fine.
There is a gotcha though. Some OEMs don't provide a windows disc at all, but rather a customised restore disc; this uses ghost or the like to clone a disc image onto the hard-drive, rather than doing an actual windows 7 install, and also has all the drivers, and additional software like the trial version of norton. These usually have additional checks as part of the clone, so will only install on the original hardware.
So have a look at what you got with your vista home pc; did you get a windows vista install disc and oem sticker, or did you get some custom restore disc that you may have had to burn yourself from files on the hard-drive?
If it's the latter, that's what they may supply for the windows 7 upgrade, and it probably won't work on anything other than the computer it's meant for.
That said, you should still get an OEM COA sticker with a key, so you might be able to use that key with a vanilla 7 home premium you acquire by other means.