My federal return cannot be filed because of zero values (no income, no refund, etc). How can I file my state return? The online software is caught in a loop.

If there is nothing on your federal return, you are not required to file.  And generally, you do not have to file a state return unless you are required to file a federal return.  What state income do you have that is not on your federal?  Note - you can't add anything directly to your state return.  Everything gets entered on the federal return and TurboTax will transfer it to your state return.  

I don't have any additional state income, I have a refund due to a credit on MO return. (answering now on mother's account - doing her taxes). It should let me e-file whichever I want to e-file, and if the state requires e-filing federal to e-file state then I'd think I should be able to do both.

Try entering a dollar of interest income on the federal return - that should allow both to be filed.

You can print and mail state.  Is it NY?  If the program doesn't let you efile Federal then it's ok to mail NY or any state return.  If you want to efile Federal try adding a fake $1 of bank interest.  Put the amount in box 1.