Nick here from Shopify.
This is a great question. The short answer is, yes, an order can be completed without a payment gateway being set up within the Shopify admin yet. Let me go into more detail about this below.
For example, the most common one we get is that Shopify Payments is added, but set up is not complete. So orders can be taken but there is nowhere to send the money too. In a case like this case, if the store owner doesn't finish setting up Shopify payments with the correct bank details etc. Shopify will refund the customers their money after a certain amount of time. However, if the payment gateway is set up in the appropriate time period, the order and payouts will go through and generate as normal. The above example is for Shopify Payments but if a store owner is using a third-party payment gateway to accept order payments, the store owner would have to reach out to the payment gateway's support directly.
Don't hesitate to ask, should you have any other questions about anything.
All the best, Nick
I'll be sure to bring this to the team on your behalf as a feature request for the future. Thanks again for reaching out and asking.
Kind regards, Nick