Odoo expense module used to allow employees submit their expenses; But you can use the below module if you want to link the supplier bill with expense line.

Supplier invoices on HR expenses | Odoo Apps Store

This module should be used when a supplier invoice is paid by an employee. It allows to set a supplier invoice for each expense line, adding the corresponding journal items to transfer the debt to the employee.

If you want to add vendor name just for reference or reporting wise you can add a new many2One field (res.partner) to the expense model and show it in expense form to be able to select the vendor.