If you are asking about making a booking not cruising to/from US ports, you should be able to. I live in Florida and cruised in September on Discovery from Port Canaveral. Currently, I have a cruise booked on Discovery from Barbados next March. Both booked online thru TUI website.
As I book cruise only, the link to that category can be found following the three vertical bars on the left of the TUI website. Look for 'cruise type' link.
Not all credit cards will be accepted for some reason or other. Did not like my US backed CC.
I use a CC which is backed by Barclays bank. You can use Paypal but there appears to be a fee to use that process.
Finally, this problem seems to be my internet provider problem. I get 'access denied' when trying to access TUI on my laptop. Works fine under cellular on my IPAD and using wifi from another internet
provider. Just mentioning this as it may have happened to others.
