Occasionally a local reaction to injectables can show up as fur loss (I'm not certain if the fur/hair loss is at the vaccine site, you did not mention a specific region), and other times we see a more severe reaction.
If there is a severe enough trigger, such as shock--a type of vascular collapse--then skin can manifest a myriad number of changes, fur loss being one of them. The fur should grow back over the next three to six months if there is no other underlying condition, and I would use a water based hypo-allergenic skin emollient to address the scaling problem.
See your veterinarian if any other changes.
Dr. Van Lienden
Dr. Raymond Van Lienden DVM
The Animal Clinic of Clifton
12702 Chapel Road, Clifton
Virginia, U.S.A. 20124