What would you call a person who possesses a certificate?

Technically, this would be dramatically overstating the certified person's achievements, but within the context of your application or website, you could reasonably refer to them as graduates:

1: a holder of an academic degree or diploma
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In this case, the "diploma" is merely the certificate that you issued, but that should be clear enough from the context of the application.

Certificated. As obvious as it seems, that is the word used, for example, for teachers in the U.S. who have earned their teacher's credentials.

Certificated: "a document attesting to the fact that a person has completed an educational course, issued either by an institution not authorized to grant diplomas, or to a student not qualifying for a diploma." Dic.com

At the risk of stating the obvious...why not "certified"? If you're licensed to teach a course of say, physical therapy, then a person successfully completing that course would be afforded the title of "Certified Physical Therapist".

The answer would really depend on the authorizing organization’s perspective on their authority. Based on what has been said so far, I would prefer the word ”certificant” or to use the word ”certificated” as a descriptive adjective before the title of the person being certificated. As pointed out by other posters, certified, even when not being used to mean crazy, has a different connotation than certificated.

Many governing bodies make a definite distinction when it comes to their documents of authorization. The organization with whom I am most familiar is the US FAA. The FAA does not issue airman licenses. They issue certificates for pilot, instructor, examiner, technician, mechanic, etc. When describing an airman, the FAA and its representative will use the term certificated (pilot, et. al.) as an adjective. Wikipedia gives the reason as, “U.S. pilots are certified, not licensed, although the word license is still commonly used informally.1 Legally, pilot certificates can be revoked by administrative action, whereas licensing (e.g., a driver's license) requires intervention by the judiciary system.”