It really depends whether you source standard parts (available from catalogue) or they make parts from your tool on your demand. If you mean injection moulding process, I would start with questions - on which injection moulding machines are the parts in question produced? You would like to know whether the machinery and tools are adequately maintained, so check for maintenance plan and records, machine / tool logbooks if they exist. Then the raw material - if these are standard parts, how much regrind is allowed and if it is allowed at all, ask for documented directions how much regrind can be used. The process itself - who releases the first good piece from production order and how it is recorded? Visual control - do they compare the controlled parts with any reference parts and if so, are these reference parts approved? If part colour is a controlled feature, how do they check that (it can be tricky if they dose and mix the pigment on IMM or, even more tricky, in raw material feeding system). Ask for records of non-compliant parts - it is normal in injection moulding that you get deformed parts in the start-up phase, and then the process can go awry too. If they claim their process produces only good parts - ekhm, that would make me suspicious. This can mean that there is no control during the process. Ask for batch release process. Then check how they make sure each packaging item contains good amount of pieces (can be tricky for small details). Assess whether their packaging procedure protects the item from damage durin transportation, in house and external. You might be interested in their plastic waste management too.
Sorry, I don't have any template. But these are questions you may ask.