PORT TALBOT, Wales—When the telltale rash appeared behind Aleshia Jenkins's ears, her grandmother knew exactly what caused it: a decision she'd made 15 years earlier.

Ms. Jenkins was an infant in 1998, when this region of southwest Wales was a hotbed of resistance to a vaccine for measles, mumps and rubella. Many here refused the vaccine for their children after a British doctor, Andrew Wakefield, suggested it might cause autism and a local newspaper heavily covered the fears. Resistance continued even after the autism link...