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Hi, I can help you with your question today.
Based on your description, I am suspicious of a few different possibilities. Toby may have a skin infection, allergy, or hypothyroidism, for example.
I recommend having him seen by your local vet. I would start with a physical exam, skin scraping (to rule out parasites), skin cytology (to look for a yeast or bacterial infection) and some blood testing to check his thyroid.
In the meantime, you can also try giving a fish oil supplement to help with general dryness as well as skin and coat condition.
It is unlikely to be related to his vaccine.
Please let me know if you have further questions. I am happy to continue until I provide the service that you seek.