Most W-2s do not include any local tax information because most localities do not have any local income tax. Here in Florida where we so not have a state income tax many W-2s do not even have any state tax information.
The only situation that not having the information would be if you needed to file a local or state tax return and it relied on some missing information.
Also, one level of government will not look to the other information. That is, the federal will not be concerned what is or is not in the state and local boxes.
Your W-2 is fine without local tax information (unless you were trying to file a local tax return).
I hope this helps.