Yes, any Dell Vista DVD will work with your Dell PC. Try to use one that has Service Pack 3 (SP3), because there will be no more upgrades from Microsoft and this is the latest. Be aware, you don't get a great browser (IE8 is no longer OK with many secure sites, nor with Google Chrome). And you don't get a current Windows Defender, so use a third party virus checker.
ACTIVATION KEY: When you re-install, you will be asked to re-enter the key. If your key works, fine. It may balk a bit, asking you to activate by Phone. That's OK, bit of a nuisance and waste of time, but usually you get activated.
In some rare cases, Microsoft may have deactivated your key. They do this if the key has been used more than the maximum number of times to re-activate, or if they flag the key as invalid, or fraudulent, or not for your geographical area (international English, for instance, sometimes won't work in US). These are some of the apparent reasons I've heard why a key is no good. Microsoft is not selling these any more. You have to buy one on ebay from a decommissioned PC, but that will generally work. Fortunately, it's not expensive and can be delivered by email, so you can get back to work soon.
You probably know this, but I'll say it anyway. Vista works OK, but Vista is not secure due to the lack of upgrades, Many sites exclude the IE8 browser, and Microsoft won't upgrade that. This lack of software support is a good reason to move on to Windows 7 or 10. You will have support and these later versions are more secure. If you are re-installing anyway, consider shifting to a later OS.