I cannot find a W2 after I filed my federal return. Is there a way to recover the data entered when adding the W2s? Specifically, I'm looking to find local tax data.

Can you please clarify what is meant by "local tax data".  What exactly are you needing assistance with?

I'm looking for the values that were entered in from the W2, boxes 18, 19, and 20.  The local income tax values.  

You can download your return and review that for the figures.  This Turbo Tax FAQ will show you how to get a copy of your W-2. https://ttlc.intuit.com/community/forms/help/how-can-i-get-a-copy-of-my-w-2/00/26328

This Turbo Tax FAQ will show you how to save your return as a PDF and print. https://ttlc.intuit.com/community/saving/help/how-do-i-save-my-return-as-a-pdf-in-the-turbotax-software-for-mac/00/26233