Yes. National banks are private businesses. No Federal banking law or regulation requires a national bank to open an account.
The best way is to start exploring is through “America’s Best Rates” on Moneyrate.com. This website will give you a list of banks that offer consistent high yields. Then go to Bankrate.com to see if any of those banks are around your area. This website also allow you to compare all the banks within your city limits along with the interest rate they offer.
Many banks will consider their checking account to be free but require that you have a minimum balance amount in order to avoid fees.