We use a software program that creates a supplier portal where the suppliers are alerted when a document comes up for renewal. The supplier then goes in and updates the document and I review and approve it.
This works great IF the supplier actually uses the portal. If they don't/wont, the software also fires off an alert 10 days before the document expires or needs to be reviewed to a person we have that is sort of a hybrid QA/Procurement admin person. That person then emails the supplier and requests the new document.
I really wish more suppliers would use things like supplier portals, because they completely automate the tasks. Unfortunately, that's not the case.
Unless you are a big player like Bimbo or Mondelez and REQUIRE your suppliers to participate in a portal or other solution like Trace Gains or whatever, you are pretty much out of luck.
For new suppliers, I provide Purchasing with a list of documents that we require that is fairly specific as to what information the document must address. For example:
Allergen Statement: Attach a document that summarizes what allergens are processed in your facility, and your Preventive Controls to eliminate cross contact.
If there are no allergens in your facility, attach a document stating that no allergens are present in the facility. This document is Mandatory.
To answer the OP question, it really depends on what your manpower situation is. IMO, ideally it should be Purchasing that performs this function, simply because they have the direct relationship with the supplier and should have the contact information for the people that they need to request the documents from.
If QA does it, then they have to rely on Purchasing to update them with current contact info.
I find this to be one of the hardest things to manage, simply because you are at the mercy of the supplier. It does not matter who "runs" it, if the supplier is not quick to turn around your request for documents, you are pretty much out of luck.
Then again really important things like GFSI certs can usually be obtained directly From the BRCGS, SQF or FSSC22000 sites.