The Hoarder badge is for "Collecting 1000 items"
This is incredibly self-explanatory. Anything you can pick up and add to your hotbar counts as an item, documents and posters DO NOT count.
There is no real way to speed the collecting up other than learning trial spawn patterns and doing them repeatedly.
Also two handy things to note:
1. The in-game badge menu tracks your progress.
2. Leaving a trial after gaining progress still counts towards it.