On the APDT trainer search page, you can limit your search to professional members who are credentialed and certified. Just click the appropriate box or use the advanced search function to narrow your search. At the $150.00 professional level, a trainer must prove certification to become a member. Members at the $100.00 level may be waiting to test for their certification. You can always ask.
Here is the
list of certifying bodies and the abbreviations they use along with an explanation. There is a wealth of information on other pages, also.
There are APDT members (at least in my area) who teach at Pet Smart. They may not note that in their listing. Pet Smart has their own accreditation program. APDT requires an independent organization to assess skills and knowledge. Some trainers have several credentials.
There are clicker trainers who are APDT members. However, you asked for kindergarten classes not a training method, so I assume you are primarily looking for socialization, as well as basic manners. APDT promotes science based, positive reinforcement training in a variety of settings and methods.
As with anyone you hire, certified or not, you should go watch a class and/or interview the trainer and/or students before signing up.