I worked in Delaware and would I be required to include this W2 in my local city taxes if I live in Cuyahoga Falls, Oh

Yes, you have to enter every W2 that you received in 2023.  The IRS and State matches their copy with what you claimed.

Every Ohio resident and every part-year resident is subject to the Ohio income tax. 

The City of Cuyahoga Falls has partnered with the Regional Income Tax Agency (RITA) for processing all tax filings. 

Non-Residents – File a tax return if you have any gross income during the tax year from sources in Delaware. 

After you finish your federal return, you'll automatically move to the State tab, where you'll see your nonresident state(s) listed in addition to your resident state.

Tip: To ensure accurate calculations, always complete the nonresident return first if filing in multiple states because your resident state might give you a credit for any taxes paid in that situation.