One thing to remember is that most if not all companies already have an “Approved Vendor List”, or AVL, for all different products and services that your company may need. The companies on the list are audits in some form by your company periodically (ideally) so that when a project appears that needs something that list is there. If the part needed is a critical component for the completion of a product (like an actual component used in the product as opposed to something needed to make/test the prototype) then you will be using that vendor long after the project is completed and the product is launched.
Companies usually have a procedure for adding in a company to the AVL and the AVL is broken down into low medium and high; each with their own set of requirements and rigor for monitoring. In the case mentioned where the service or product is critical to the functioning of the product it would be a high and likely require an auditor to actually visit their manufacturing site (if it is a part or service site if it is a service) and audit the company to ensure they comply to necessary regulatory requirements such as ANSI standards, or IEC etc. If they are registered for complying with the specific standard then that may help the credibility and reliability of the vendor since you don’t just want a perfect product once but always.
checking if the vendor has a quality management system QMS will also help with picking the right supplier.
Companies outsource all the time and in many cases, they don’t actually “manufacture” a single component of their device and simply assemble the device components.