Finding the shovel in Sons of the Forest is a more daunting task than one might think. First you'll need to accumulate the rebreather and rope gun which can be found on these points on the map:
The rebreather can be found inside a cave on the beach and the rope gun can be found in a cave in between two rivers. Each of these caves will go all the way through so you won't have to trace your steps back. Be sure to bring a light source of some kind so you don't go in dark. You'll also have to use the rebreather to go underwater to access the cave and the rope gun to go across a zipline.
This cave is pretty linear so you should eventually come upon a giant research facility with a dead body, shovel in hand outside of it. Once you have the shovel in hand, you'll have to retrace your steps to leave the cave.