A little-known European medical team is poised to become one of the most important groups in the shifting landscape of U.S. abortion bans. Aid Access, an online-only service run by a Dutch physician, Dr. Rebecca Gomperts, began shipping abortion pills to Americans from abroad four years ago. The organization’s team consists of about four doctors supervising about 10 medical staff members, and they’re difficult for U.S. authorities to reach because all are outside the country and they ship pills from a pharmacy in India. Opponents of abortion rights have so far largely found themselves powerless to stop Aid Access from...