So after some further investigation I have discovered that the problem is more fundamental than I thought. When I remove all filters of any kind, no purchase orders show up under any grid view.
To confirm this, I created a new Vendor Portal screen that is an identical duplicate of our Purchse Order screen. When I'm logged in through the admin and visit this screen, all the purchase orders show up as expected. When I log in through the Vendor Portal through one of our vendor accounts, none of the purchase orders show up.
Here is the screen while logged into Method:
Here is the same screen while logged in as a Portal user (again, there are no filters set up):