Carly Fiorina plans to launch her presidential campaign on May 4, in an online announcement that dispenses with the pageantry that has become de rigueur in 2016 White House runs.

Instead, Mrs. Fiorina, the former chief executive of Hewlett-Packard Co., will formally declare her Republican campaign online and hold a conference call for the national press, according to a person with knowledge of the campaign’s plans.

The...