Can a wizard use the spell Levitate on a target and shoot it with attacking spells that don't require concentration or can somebody else shoot the target?
Both the wizard and other people can attack the target
Levitate has no special ending conditions for the target being attacked (the text description of levitate would have to say so). Further, a wizard can keep concentrating on a spell and cast other spells that don't require concentration (see this exact question). This means that casting levitate on a target and then having people ranged attack the target is a perfectly valid strategy.
This is actually a common strategy to employ against enemies that have no ranged attack options (do note that unwilling targets get a single Constitution saving throw to avoid the effect when the spell is cast). Barring some outside help the target is usually doomed.