While Shippingport Atomic Power Station in Pennsylvania was the first full-scale nuclear plant to be designed and built to provide commercial electric power, the Vallecitos Atomic Electric Power Plant in California was actually the first privately-financed nuclear power plant to provide meaningful amounts of electricity for public use.
Vallecitos was mainly a training and test facility, paving the way for the full-scale Dresden-1 nuclear plant in Illinois and many others that followed. Vallecitos achieved criticality in August 1957, and connection to the local utility electric grid began that October.
Vallecitos received Power Reactor License No. 1 from the Atomic Energy Commission.
Two highly informative sources on the history of Vallecitos:
Looking Back: Vallecitos Atomic Power Review
An International Mechanical Engineering Landmark: The Vallecitos Boiling Water Reactor American Society of Mechanical Engineers/Mt. Diablo Section
Thanks to Jeff Quitney