You have likely missed some required steps from earlier in the game. Here's how to do the dog quest:
You need to have used the dog dido a few times, befriended the dog, talked to the park bench vet guy, and fed the dog with dog treats a couple of times. Buddie will then be in pain. The park bench guy will ask you for a stethoscope.
To get the stethoscope you must have done the first Dominic scene, and have the doctor's unlocked, you'll get it from Dominic at the end of the scene. If the doctor is unavailable, you'll get called for a visit after 60 enchantments. Once you have the stethoscope, return to the park bench guy, play on from there.
