Once the player arrives on Donar Island for the main story, they also happen to unlock Thunderous Cave. In order to get to the area with Lunarpup, the player must first traverse through the main story and unlock the worker helmet. This helmet will light up the darker areas of the cave, one of which is where you can find the popular Coromon, Lunarpup.
Once you enter the cave, you need to take the path to the right. If you go straight up afterwards, you will find a chest. After taking the chest, follow the path until you hit a cross road with 4 ways to go.
After fighting the trainer, the player will find themselves with another choice of going right or left. Go right. Follow the path until you reach this point.
Head up the stairs, and fight the last trainer blocking your way before reaching Lunarpup. After beating that trainer, there will be a ladder up those stairs. Climb down it to enter a dark area with a chest. Run around in the dark area, and you can find a Lunarpup. There is another dark area in the cave, but Lunarpup does not spawn there. They can only spawn in this dark section of the cave.