Is there any correlation between immunizations and eczema in children?

Not really: Unless there is a specific component within the vaccine that your child is allergic to, then vaccines should not cause flare up of eczema. Your child is exposed to numerous allergens every single day, and it's likely these are the triggers. Work with an allergist to determine what your child's triggers are, and talk with your pediatrician about maximizing skin care to avoid flare-ups of the eczema.