Eunice here from Shopify, thanks for posting, and welcome to the Community!
You're able to send invoices to customers directly by creating a draft order. The draft order is converted to an order once payment has been accepted. You can accept payment by sending a secure checkout link, or by confirming you've received payment elsewhere such as over the phone or in person. To learn more about draft orders and how to create one check out this link.
To ensure I am giving you the best advice, I would just like to learn a little more about the order in question. Has the customer told you their PayPal account has been charged for the order but you can't see the order on your orders page?
I look forward to hearing from you.