Here are some ideas :

• Vitamin A, vitamin k2, copper, B-complex from beef/calf liver (see chart below *)
• Vitamin C citrus fruits (if tolerated), berries and vegetables (sauerkraut has some too).
• Vitamin E from red palm oil
• Niacin from tuna, calf's liver, ...
• Chromium from a decent amount of vegetables and onions if tolerated (dulse is a good source too, but probably not tolerated). Brewer's yeast contains this too, though you probably consider that a supplement.
• Potassium from a decent amount of vegetables (most vegetables are a good source, dulse is a great source but probably not tolerated)
• Selenium from fish and beef liver.
• Magnesium from mineral water, mustard greens, swiss chard, halibut, spinach (and supplementation)
• Calcium from egg shells, bone meal, bone broth, ...
• Gelatin from bone broth or supplementation
• Zinc from red meat, oysters, ...
• Iron from red meat, though excess iron should be avoided. Don't eat only muscle meat, eat gelatin and organs.
• Phosphorus vegetables, beef liver, ... contain adequate amounts.
• Manganese from vegetables (spinach!)
• Iodine from wild salmon and other fish
• Sulfur from broccoli, onions, garlic (only eat FODMAPS if tolerated), meats, ...
• Probiotics from fermented foods.
• Choline from beef liver and eggs (if tolerated)
• CoQ10 from beef heart
• Anti-inflammatory compounds from turmeric and lots of other paleo foods
• Anti-oxidants from berries
• Omega-3 from salmon (wild), mackerel, ...
• Saturated Fats from coconut oil, butter, ...

I would still supplement fat-soluble vitamins and magnesium. Don't forget whole cod liver is a great source of those fat-soluble vitamins!

Most of these compounds are beneficial to the skin. Still, there is more to acne : don't wash your face with harsh stuff. Drink apple cider vinegar if it helps you. Use smart topicals. Don't be afraid of fruits and aloe vera. Experiment.

Mark's Daily Apple made a good article about acne too. He mentions that stress and sleep are very important. I hope you will agree. That's why I believe magnesium should still be supplemented, and vitamin D too (unless you're in the sun all the time). I believe some people on here had success with magnesium threonate taken before bed.

That being said, I hope you like liver ;)

*this is the nutritional composition of 100g raw beef liver. Needless to say it's a must!

So zinc did wonders for my acne -- apparently raw oysters are a good source. You can also get it from sunflower seeds, peanuts, and lamb. I take the tablets, m'self, but good luck with your quest.

From an acne sufferers perspective liver should be the mainstay, I believe ~750g/week really helped my skin. It's probably the vit A but just about everything else is in there to so who knows?

As for milk; take it slowly and try to get the best quality possible. I use raw organic milk and lots of it but if I get cheeses from less reliable sources I break out a little. the lesser quality milk is still grass fed in the UK so I'm unsure what component is causing the skin flare up.