Alternators often are their own tensioners; they sometimes mount on a hinged bracket and a slotted bracket, so the alternator turns on the bracket to tension the belt, and then the bolts are tightened down all the way. Depends on the car though, there may be an accessory tensioner somewhere else.

Check for a bad grounds or corroded cables/connections before you blame the alternator. If you checked voltage at the battery, you're also including the cables and ground points in your diagnosis, so make sure you fully understand what you're looking at.