Two puppies, described as Staffy-types and around four months old, were left in a box at the West Mount Vets on Pellon Lane, Halifax, at 5.30pm on February 7 by a man dressed in a hoodie and tracksuit - who left before staff realised what was happening, the RSPCA said.
A week later, at 5pm on February 14, a third puppy was left at the same vets, with investigators suspecting the puppies were all left by the same man.
The puppies' condition is caused by a severe case of a skin condition called demodectic mange, the charity said - the red, itchy skin condition is caused by mites, which can lead to infections and hair loss if not treated properly.
RSPCA inspector Sarah Bagley said: "It's unbelievable that someone has dumped these puppies in this way, not once but twice.
"It's taken a long time for the mange to take hold like this, in fact it's thought these puppies have probably had it for most of their lives."
The puppies were taken to RSPCA Leeds, Wakefield & District Branch Animal Centre, where they are on daily medication and having regular medicated baths to help manage their fur loss.
Now on the road to recovery, the trio having settled into their new lives with new names to match - Albus, Winky and Dobby, after the Harry Potter characters.
Gena Cameron, branch administrator, said: "I'm so happy to say that they're improving every day.
"They were really nervous when they arrived and shaking.
"Their skin was so red and angry-looking when they first got here, but their fur has already started growing back at the base of their tails.
"It'll be some time until we're able to rehome them, but things are definitely going in the right direction."
Ms Bagley added she was "very concerned" about other animals the suspect may have in their care, given the severity of the puppies' condition.
She urged anyone with information to call the appeal line 0300 123 8018.