No. When the total is zero there is nothing to pay and nothing else to do. Complete Order finishes the order on its own: no card, no payment provider, no currency step. Do not buy it a second time.
Why the button changed

The cart swaps Proceed to Checkout for Complete Order the moment the total reaches exactly zero. It is the same button doing a different job. Instead of handing you to a payment provider, it records a completed order for nothing.
What pressing it does
- The order is recorded and marked successful straight away.
- You land on the processing page, which reads "Payment Successful!".
- Whatever you bought is applied exactly as a paid order would be, so a competition entry is created and appears on your competitions page.
- The order shows in your transaction history with its payment method given as Free/Membership.
There is no provider receipt, because no provider was involved. If you need a document, open the order from your transaction history and use Download Receipt on the detail page.
What actually makes a total zero
- The thing is free to begin with. A free competition is the common case.
- A discount covers the whole amount. Discounts can be a percentage or a fixed sum, and either can land the total on zero.
Using a membership entry is not this. That is a separate path with its own confirmation, and it does not produce a zero total cart. See Registering with a membership entry.
A cart with anything payable in it cannot use this
If any item in the cart still costs money, the zero path is refused outright: "Cart contains paid items. Please use the regular payment flow." So a cart that offers you Complete Order genuinely owes nothing.
If you are still unsure it worked
Check the entry rather than the payment. Open your competitions page and look for the registration. If it is there, the order went through, whatever the amount was.