Unfortunately, we've had lots of experience with black spots on dogs that we've rescued. It's from a systemic yeast (candida) infection. The best thing to do is get the dog on a raw meat and bones diet immediately, add probiotics, digestive enzymes, coconut oil, a bit of raw goat milk keifer or yogurt daily and a small amount of raw green vegetable like organic spinach or brocoli. NO kibble, pet foods, starches, carbs, sugar and other junk. These will only enable the yeast to live and grow. You want to feed foods that kill the yeast.
We also added 1 capsule of Seagate olive leaf extract to their meal every day; it's a powerful fungus killer and also strengthens their immune system. The last dog we got who had this had a really bad case. She even smelled yeasty after getting baths. Doing the above, she was clear of yeast in just 2 months; no more itching, red skin, smell and black spots. Her hair also grew in nice and full; it was at least twice as much as she had when she was full of yeast. That was 2 years ago. She's been healthy ever since on her raw meat/bones diet.
Oh, and stop the shots. Vaccines worsen any health problem and also disrupt the immune system. They cause way more problems and don't help like we've all been told.