No, they don’t need a vaccination card because the state of California already maintains a registry within the Department of Health. The state health department is assigning a random number to each person within the vaccine registry and then handing those numbers over to the state lottery agency who will then conduct the drawings to determine prize winners. The selected numbers will be given to the health department, who will then contact the individuals who’ve won.
Did you not read the news story about this? All the details are within, or within links from the news article. They don’t need a vaccine card because they have the data necessary to conduct the lottery.