A short guide on where to find the shovel from the Darktide Collaboration DLC
Location
You can find the shovel during chapter 1 on Rykad Minoris. It is in the northeast corner of the Streets of the Capital in the Upperway.

Investigate this area.
Weapon Stats

Conclusion
If you already completed chapter 1, you should receive the shovel during a separate event in the captain’s quarters.