There's no specs on how snug the chain should be, nor is there a way to adjust the slack.
It only drives the oil pump, so it's not like it's under a huge load or anything.
It's possible for the chain to wear out (just as any part), but since it's bathed in filtered oil, it's got to last practically forever.
I wouldn't worry about it. Unless something caused the chain and sprockets to wear, or were damaged by loose bits or debris.