Cannot get Turbo Tax system to accept credit card payment. Tried multiple cards

I'm not sure if this will help in your case, but try disabling your main internet browsers pop up blocker before getting to the payment page.

Also, make sure the address on your return matches the billing address for your card.

You need to call support and tell them there is a billing problem.  Most of them will not know what to do so tell them you need them to contact billing.  You need to call Monday thru Friday between 9:00 AM and 4:00 PM Mountain Time.  They can look at  your account and fix what is causing the problem.