I have a vintage road bike, at the moment there are loose metal bearings in the hubs. Some of the bearings are missing. Can I buy cartridge bearings or do I need to buy loose bearings?
To use cartridge bearings, the hubs must have been designed to use them (absurd kludges aside).
Your hubs will almost surely be classic cup-and-cone hubs, and those will need replacement bearings.
However, do note that the proper number of bearings for a hub is not necessarily a number that completely 'fills' the cup bearing race. You might not be missing any. 9 is typically common, with a little bit of space where it looks as though you could fit a 10th or 11th. Check with your LBS for how many you'll need for your particular hub.
Generally when new there is room for about half a ball additional in the bearing race, but very quickly wear of the cup makes it look like a ball is missing. And a ball bearing assembly can function reasonably well with 2-3 balls actually missing (though this will cause faster wear).
The usual technique, when you don't know for sure if any balls have been lost (or how many) is to go for "full minus one" -- enough balls to fill the race, then take one out.
The balls are cheap and any decent bike shop will have them. (Take a ball or two with you for the shop to measure.) If you replace any balls, though, replace all of them, since the old ones will be worn and not have exactly the same diameter. (And always get 2-3 more than you need, since you'll always drop one or two and not be able to find them.)
I've found that normally there is an odd number of balls -- 9 or 11, eg. But there's no theoretical basis for this rule.
To use cartridge bearings you'd have to replace the hubs, or at the very least get a machine shop to machine the hubs to accept the cartridges (thought there's likely not enough "meat" to allow this).
I've had cartridge hubs in the past and they're not necessarily trouble-free. Cartridges were "the in thing" back maybe 1985, but now you only see them in cranks (where there is room to place a cartridge with no compromises). But if you want cartridge hubs, I believe Phil Woods is still the go-to company to get them (or was last I checked). (And note that the cartridge hubs will not look like standard hubs.)
If you are buying new, then sealed would be a great way to go because there will be almost zero maintenance and the free roll is not comparable. Loose bearings can do very well but may require periodic maintenance for normal road use and more maintenance if ridden in dirt or weather.
Either one depends on how you setup and mount the wheel. Most people tighten hubs too tight, thus binding the bearings and that defeats the purpose. If you snug up the wheel hub nut, mount the wheel and tighten the wheel nut. Almost ALWAYS, the hub nut will get a little extra tension on the bearing and then the bearing is in a bind and does not roll as well and can even not free roll at all.
YOU SHOULD snug up the hub nut and turn it back a quarter turn on one side only (becsue the axle is free between these two nuts with the hub in the middle). Mount the wheel and check the roll out on the wheel. feel if there is any binding (or slack) on the bearings.
Binding will cause premature wear and you WILL have to replace them eventually. This goes for sealed cartridge bearings as well!
If they do not roll freely or you feel binding, loosen the wheel and loosen the hub nut 1/8 turn at a time and tighten wheel again. You want to have the bearing just snug but no slack! That will be the sweet spot when doing either bearings. If it is too loose, the you can wiggle the wheel and feel it. Tighten it up just a bit same as described above but opposite direction.
Bearings will wear if you ride a lot. Sealed bearings will last a lifetime of the wheel for most average riders just keep the dust and dirt off them.
Have fun and ride Rad!