I'm sorry to hear about your puppy's allergies, Claire. Allergies can be one of the toughest things to diagnose. My Flurry, who will be 4 in July, started with skin allergies when she was about 8 months old. She had small lesions on her skin that obviously itched and bothered her. My vet ran all kinds of tests and everything came up negative, except that the lesions always tested positive for staph. After a year of having her on and off antibiotics, I took her to a canine dermatologist with my vet's blessing. The derm doc found that Flurry is actually allergic to the staph her own body produces. She is, essentially, allergic to herself.
I make her a home-cooked diet with a recipe from the experts at Natures' Farmacy; she gets probiotics and a dermal support supplement, and I give her allergy injections once a week with staph serum. She has occasional flareups which we treat with antibiotics. She has never grown undercoat, which I attribute to her skin allergy. It's been a long haul, but at least we know what's going on and how to treat it.
This is a long way of suggesting that you take your puppy to a canine dermatologist. I love my vet, and they're terrific, but a dermatologist will perform specialized tests that your vet doesn't, and can pinpoint the source of the problem and the solution, or treatment, much faster. It's worth the effort and money. I wish I'd done it sooner! Allergies don't resolve on their own, and can get worse. Your vet will give you topical treatments or meds but until you find the cause, it won't solve the problem.
I wish you the best of luck with this problem, and hope your puppy improves soon!