I assume that this is a private company and a not a government agency. The law does not specify any particular process that a private company must use to decide promotions. So no law says that a company must use an application process for deciding promotions. The company can just have a manager select the person he or she thinks is the best for the job.
What federal and California law do say is that the company cannot engage in illegal discrimination. Making the promotion decisions based on race, color, national origin, or citizenship would be illegal discrimination. So the question here is whether the company decided the promotions based on those factors or based on something else. The fact that 4 of the 5 promotions were given to nonwhite employees tells me nothing about whether race, color, national origin, or citizenship were the reasons for the promotion choice. It could have been that those persons were simply the best ones for the job. You'd need to dig a bit more to determine if illegal discrimination was at work here.
In addition to the questions quincy asked, how many people were eligible for the promotions and out of that pool, what percentage were white?