From all my research I do not believe that this story is true.
It is true that many of the Native Americans died during smallpox epidemics/ This disease was introduced by the Europeans. Recent mission scholarship is also critical of the crowded conditions in the mission housing and the fatalistic attitude of the Spanish to these outbreaks and to the rejection of innovations in medicine that could have saved lives. Dr. Robert H. Jackson, in his book Indians, Franciscans, and Spanish Colonization sites the experience of the Baja missions where only three missionaries used inoculation by variolation, which substantially reduced the mortality rate at their missions. This book offers a scholarly perspective on the treatment of the Native Americans in California.