Amazon.com Inc.’s foray into the pharmacy business is causing the company to clash with entrenched industry incumbents that are putting roadblocks in front of the company’s growth plans.

Last week, Surescripts LLC, a provider of the technology widely used to route electronic prescriptions, accused Amazon’s mail-order pharmacy subsidiary PillPack of receiving patient data that it had fraudulently obtained through a third party. Surescripts went public with the allegations in a news release.

PillPack,...