Simon and Schuster (book publishing house) has openings for paid summer internships.
Simon and Schuster also has a Children’s book cover designer position open https://cbs.avature.net/cbssscareers/JobDetail/Children-s-Book-Cover-Designer/12708
Viz Media (One Punch Man, Sailor Moon, etc.) has a designer position open http://jobs.jobvite.com/viz/job/ocNJ8fwA
Candlewick Press has an opening for an associate designer and also has several internship openings
Several internships available for graphic designers for Pro Sports teams through TeamWork Online