How do I get a copy of the form I need to mail in to my state as proof that I paid state tax online?

If you mean something from TurboTax, we have nothing because TurboTax doesn't pay your tax.  All you do is enter how you want to pay your state tax and that's transmitted to the state but it's up to the state to withdraw money from your bank or to charge your credit card.  You'd need a copy of a bank statement or credit card statement showing you paid.  If you paid them directly online somehow you'd need a copy of that payment from the state.

if you paid online it will register on the detail statement for the instrument you used to make the payment.

If it does not, then the State is correct.