Ash here from the Shopify team. Welcome to the Shopify Community!
This definitely sounds like a strange issue, and I'm happy to help out! The first thing I recommend is working with this customer to confirm if they indeed paid for their order. I suggest reaching out to the customer and finding out:
Once you know the payment method used, and if it appears that they were charged, you can check that payment method on your end to see if payment was received. If it turns out payment was received but no order was created in Shopify, that would likely be the result of a third-party payment provider being used, and the customer not waiting to be redirected back to Shopify after completing payment. In order for our system to create an order, the customer needs to reach the Shopify "thank you" page. If this is the case, you can create a Draft Order for the customer, add the products they ordered, mark it as paid, and fulfill it normally.
I hope this helps, and feel free to let me know what happens when you get some more information from your customer! I'll be happy to continue assisting.