I have a client that wants a user / customer to have the ability to look up an order number on the home page and pay for it without setting up a customer account.
Example: • Customer goes to the home page of the website • Customer locates a field with "Search Order Number" above it • They enter the order number and see order details and cost • They pay for the order
My client would manually set up the orders and then give the customer the ability to come in when they want and pay. I understand an email can be sent out to finish and pay for the order but this would be an alternative method.
There's a lack of detail here that makes it vague as to what would be actually needed.
Overall though this is very advanced customization, or app, or both.
Your asking to expose backend potentially private information to the frontend without any authentication of random visitors.
In almost all cases to maintain privacy unique unguessable links should be used, not arbitrary search exposing information (see access apps like locksmith).
Generally in all cases customers will go through checkout and fill in info in addition to payment.
If these "orders" are just glorified carts that need to be prefilled use permalinks along with pages, or even blog articles, or as products (i.e. /products/order-001) as though will show in search( or a custom search constrained to specific resource types)
If these are draft orders that the merchants staff creates you'll may need an automation app like mechanic , to possibly duplicate draft-orders to provide checkout links to each new possible visitor or aid in sending different emails.
If you need consultation, or know the customization or automation you need then contact me directly by mail for services. Please always provide context, examples: store url, theme name, post url(s) , or any further detail. Contact Info in signature.