QuickBooks doesn't allow you to enter the same name for a customer and a vendor. You can consider adding a unique character that distinguishes one name from the other.
For example, your supplier's name is Mark and you want to add a new customer with the same name, you can enter a number or a character that identifies the name, like Mark 1. This way, you'll be able to create a new customer.
Once done, you can now create an invoice for your new customer. For more guidance, feel free to watch this video to learn how to create invoices: How to Create a Customer Invoice.