Americans at high risk from COVID-19 because of severely weakened immune systems are now allowed to get a third vaccination in hopes of better protection, a policy change endorsed Friday by influential government advisers.

The Food and Drug Administration ruled that transplant recipients and other similarly immune-compromised patients can get a third dose of either the Pfizer or Moderna vaccine. But the decision offers an extra dose only to those high-risk groups — not the general public.

These patients have been clamoring for months for better protection, some of them traveling across state l...