Welcome and thank you for asking about putting a field for an order number. I see that this is very important for your records, so I'll be sure to help you out.
If you're using QuickBooks Online Plus, you can add a custom field and name it as Order Number, so you can enter such info when you create invoices. If you're on Essentials or Simple Start, just enter the number in the Message on invoice box, as a workaround.
Go to Settings ⚙, then select Custom fields.
Select Add field.
Enter a name in the Name field as Order Number or anything you like.
Select the All Sales forms checkbox.
If you want the custom field to appear on printed and delivered forms, turn on the Print on form switch. Otherwise, it will only be visible in QuickBooks.