Probably not. I know there are on-line testimonials praising the therapeutic value of vinegar. In my experience it makes you smell like a salad, and that's about it. There are plenty of legitimate eczema treatments available. Your dermatologist can help you choose one that is appropriate for you.
Yes. Secondary infection with bacteria especially staph is common with eczema. Vinegar diluted in bath water serves as an antimicrobial and helps control eczema flares. Studies have shown that most children are infected with staph which leads to eczema flares. The vinegar needs to be really diluted (about 1/2 cup in a bath tub). Don't ever pour vinegar directly on the skin.